Answer:
[tex]P(16.6 < \bar X < 22.6) = P(\frac{16.6-19}{3} <Z< \frac{22.6-19}{3})= P(-0.8 < Z < 1.2)[/tex]
[tex]P(16.6 < \bar X < 22.6) =P(-0.8<Z<1.2) = P(Z<1.2)-P(Z<-0.8) = 0.88493- 0.211855= 0.673[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(19,6)[/tex]
Where [tex]\mu=19[/tex] and [tex]\sigma=6[/tex]
And we select n =4 fish. For this case we want to find this probability:
[tex] P(16.6 < \bar x < 22.6) [/tex]
And since the distribution for X is normal then the distribution for the sample mean is also normal and given by:
[tex] \bar X \sim N(\mu , \frac{\sigma}{\sqrt{n}}=3)[/tex]
And the z score is given by:
[tex]z = \frac{\bar x -\mu}{\sigma_{\bar x}}[/tex]
And if we apply this formula we got:
[tex]P(16.6 < \bar X < 22.6) = P(\frac{16.6-19}{3} <Z< \frac{22.6-19}{3})= P(-0.8 < Z < 1.2)[/tex]
And we can find this probability with this operation using the normal standard table or excel:
[tex] =P(-0.8<Z<1.2) = P(Z<1.2)-P(Z<-0.8) = 0.88493- 0.211855= 0.673[/tex]
To find the probability that the mean weight of four randomly selected fish will be between 16.6 and 22.6 pounds, we can use the Central Limit Theorem. The probability is 0.7556.
Explanation:To find the probability that the mean weight of four randomly selected fish will be between 16.6 and 22.6 pounds, we can use the Central Limit Theorem. The Central Limit Theorem states that if we take multiple samples from a population with any distribution, the distribution of the sample means will approach a normal distribution. In this case, we have a normally distributed population with a mean of 19 pounds and a standard deviation of 6 pounds.
To calculate the probability, we need to standardize the range of weights using the formula for the standard error of the mean:
Standard error of the mean (SE) = Standard deviation / sqrt(sample size)
We will use the formula:
Z = (X - mean) / SE
Where X is the upper and lower bounds of the range, mean is the population mean, and SE is the standard error of the mean.
First, let's calculate the standard error of the mean:
SE = 6 / sqrt(4) = 3
Then, we can calculate the z-scores for the upper and lower bounds:
Z_upper = (22.6 - 19) / 3 = 1.2
Z_lower = (16.6 - 19) / 3 = -1.1333
Since the z-scores are in standard deviation units, we can look up the corresponding probabilities in the standard normal distribution table:
P(16.6 < X < 22.6) = P(-1.1333 < Z < 1.2)
Using the table, we can find the probabilities:
P(Z < -1.1333) = 0.1293
P(Z < 1.2) = 0.8849
Finally, we can calculate the probability between the two bounds:
P(16.6 < X < 22.6) = P(Z < 1.2) - P(Z < -1.1333) = 0.8849 - 0.1293 = 0.7556
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Determine algebraically whether the function is even, odd, or neither even nor odd.
f(x) = 3x^2 - 1
Neither
Even
Odd
Answer:
Given Function is an even function
Step-by-step explanation:
Explanation:-
Even function :-
A function f is even if the graph of f is symmetric with respective to the y - axis.
Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
Odd function : -
A function f is odd if the graph of f is symmetric with respective to the origin
Algebraically, f is odd if and only if f(-x) = - f(x) for all x in the domain of f.
given function is [tex]f(x) = 3 x^2-1[/tex]
[tex]f(-x) = 3 (-x)^2-1=3 x^2 -1 = f(x)[/tex]
therefore f(-x) = f(x)
given function is an even function.
A baker need 6lbs of butter for a recipe .She found 2 portions that each weigh 1 1/4 and a portion weight 2 3/4 does she have enough butter for recipe
Answer no she does not have enough:
Step-by-step explanation:
2(1.25) = 2.5
2.75
2.5+2.75 =5.25
Time to go to the store!
Answer: The baker doesn't have enough butter.
Step-by-step explanation:
Total number of pounds of butter that the baker needs for the recipe is 6 pounds.
She found 2 portions. One of them weighs 1 1/4 pounds. Converting 1 1/4 pounds to improper fraction, it becomes 5/4 pounds.
The other portion weighs 2 3/4 pounds. Converting 2 3/4 to improper fraction, it becomes 11/4 pounds.
Total amount of butter that the baker has would be
5/4 + 11/4 = 16/4 = 4 pounds.
Therefore, the baker doesn't have enough butter.
Alyssa spent $35 to purchase 12 chickens. She bought two different types of chickens. Americana chickens cost $3.75 each and Delaware chickens cost $2.50 each. Write a system of equations that can be used to determine the number of Americana chickens, A, and the number of Delaware chickens, D, she purchased. Determine algebraically how many of each type of chicken Allysa purchased. Each Americana chicken lays 2 eggs per day and each Delaware chicken lays 1 egg per day. Allysa only sells eggs by the full dozen for $2.50. Determine how much money she expects to take in at the end of the first week with her 12 chickens.
Answer:
The System of equation to determine the number of chickens purchased is [tex]\left \{ {{x+y =12} \atop {3.75x+2.5y =35}} \right.[/tex].
Alyssa purchased 4 Americana chickens and 8 Delaware chickens.
Alyssa will expect to make $23.33 at the end of first week with her 12 chickens.
Step-by-step explanation:
Given:
Let the number of Americana chickens be 'x'.
Let the number of Delaware chickens be 'y'.
Number of chickens purchased = 12
Now we know that;
Number of chickens purchased is equal to sum of the number of Americana chickens and the number of Delaware chickens.
framing in equation form we get;
[tex]x+y =12 \ \ \ \ equation\ 1[/tex]
Also Given:
Cost of Americana chickens = $3.75
Cost of Delaware chickens = $2.50
Total amount spent = $35
Now we know that;
Total amount spent is equal to sum of the number of Americana chickens multiplied by Cost of Americana chickens and the number of Delaware chickens multiplied Cost of Delaware chickens.
framing in equation form we get;
[tex]3.75x+2.5y =35 \ \ \ \ equation\ 2[/tex]
Hence The System of equation to determine the number of chickens purchased is [tex]\left \{ {{x+y =12} \atop {3.75x+2.5y =35}} \right.[/tex].
Now to find the number of each type of chickens she purchased we will solve the above equation.
First we will multiply equation 1 with 2.5 we get;
[tex]2.5(x+y)=12\times2.5\\\\2.5x.+2.5y = 30 \ \ \ \ equation \ 3[/tex]
Now we will subtract equation 3 from equation 2 we get;
[tex]3.75x+2.5y-(2.5x+2.5y)=35-30\\\\3.75x+2.5y-2.5x-2.5y=5\\\\1.25x=5[/tex]
Now Dividing both side by 1.25 we get;
[tex]\frac{1.25x}{1.25}=\frac{5}{1.25}\\\\x= 4[/tex]
Now we will substitute the value of 'x' in equation 1 we get;
[tex]x+y=12\\\\4+y=12\\\\y=12-4 = 8[/tex]
Hence Alyssa purchased 4 Americana chickens and 8 Delaware chickens.
Now Given:
Number of eggs laid by American chicken per day = 2 eggs
Number of eggs laid by Delaware chicken per day = 1 egg
Cost of 12 eggs = $2.5
Total number of days = 7
Now first we will find the Total number of eggs laid by both the chickens.
Total number of eggs laid per day = [tex]4\times2 + 8\times 1= 8 +8 =16\ eggs[/tex]
Total number of eggs laid in week = [tex]16\times7= 112[/tex] eggs
12 eggs = $2.5
112 eggs = Cost of 112 eggs.
By cross multiplication we get;
Cost of 112 eggs = [tex]\frac{2.5 \times 112}{12} = \$23.33[/tex]
Hence Alyssa will expect to make $23.33 at the end of first week with her 12 chickens.
The system of equations that can be used to determine the number of Americana chickens, A, and the number of Delaware chickens, D, she purchased are as follows;
A + D = 12
3.75A + 2.50D = 35
Alyssa purchased 4 Americans chicken and 8 Delaware chickens.
She is expected to take in $22.5 at the end of the first week with her 12 chickens.
number of Americana chickens = A
number of Delaware chickens = D
Therefore,
A + D = 12
3.75A + 2.50D = 35
A = 12 - D
3.75(12 - D) + 2.50D = 35
45 - 3.75D + 2.50D = 35
-1.25D = -10
D = -10 / -1.25
D = 8
A = 12 - 8 = 4
A = 4
Therefore, Alyssa bought 4 Americans chickens and 8 Delaware chickens.
Each American chicken lays 2 eggs per day and each Delaware chicken lays 1 egg per day.
She only sells the egg in full dozen for $2.50.
The amount of money she expects to take in at the end of the first week with her 12 chickens is calculated as follows.
1 week = 7 days
Number of American chicken eggs(first week) = 7 × 4 × 2 = 56 eggs
Number of Delaware chicken eggs(first week) = 1 × 7 × 8 = 56 eggs
Total eggs = 56 + 56 = 112 eggs.
She can only sell full dozen of eggs. Therefore,
112 / 12 = 9.333
1 dozen = $2.50
9 dozen =
cross multiply
Amount made from the eggs = 9 × 2.50 = $22.5
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A college student organization wants to start a nightclub for students under the age of 21. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. They expect that about 74% of the student body would respond favorably.(a) What sample size is required to obtain a 95% confidence interval with an approximate margin of error of 0.03?answer: 822(b) Suppose that 54% of the sample responds favorably. Calculate the margin of error for the 95% confidence interval.
Answer:
a) Sample size = 822
b) Margin of error = 0.03407
Step-by-step explanation:
We are given the following in the question:
p = 74% = 0.74
a) Sample size is required to obtain margin of error of 0.03
Formula:
[tex]\text{Margin of error} = z_{\text{statistic}}\times \sqrt{\dfrac{p(1-p)}{n}}}[/tex]
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
Putting values, we get,
[tex]0.03 = 1.96\times \sqrt{\dfrac{0.74(1-0.74)}{n}}\\\\n = (\dfrac{1.96}{0.03})^2(0.74)(1-0.74)\\\\n = 821.24 \approx 822[/tex]
Thus, the sample size must be approximately 822 to obtain a 95% confidence interval with an approximate margin of error of 0.03
b) Margin of error for the 95% confidence interval
p = 54% = 0.54
Formula:
[tex]\text{Margin of error} = z_{\text{statistic}}\times \sqrt{\dfrac{p(1-p)}{n}}}[/tex]
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
Putting values, we get,
[tex]\text{Margin of error} = 1.96\times \sqrt{\dfrac{0.54(1-0.54)}{822}}\\\\=0.03407[/tex]
The margin of error now will be 0.03407.
Based on the sampling information given, the sample size will be 822.
SamplingThe margin of error is given as 0.03. Therefore, the sampling size will be:
= (1.96/0.03)² × 0.74 × (1 - 0.74)
= 822
The margin of error for a 95% confidence interval will be:
= 1.96 × ✓0.54 × ✓0.46 × ✓822
= 0.3407
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Prove the following theorem indirectly. We will give you a start.
Prove that a triangle cannot have two right angles.
A triangle cannot have two right angles. Suppose a triangle had two right angles.
(This is Geometry)
Answer:
if a triangle had two right angles it would not be complete as to make it a triangle all corners have to meet while a 2 right angled triangle does not meet that.
i believe this is the answer
An art club wants to sell greeting cards using members drawings. Small blank cards cost $10 per box of 25. Large blank cards cost $15 per box of 20. You make a profit of $52.20 per box of small cards and $85 per box of large cars. The club can buy no more than 350 total cards and spend no more than $210. How can the art club maximize its profit
To maximize profit, the art club should find the combination of small and large greeting cards that satisfies the given constraints and generates the highest total profit. The optimal solution will provide the number of boxes of small and large cards that should be bought to maximize profit while staying within the given constraints.
Explanation:To maximize profit, the art club should find the combination of small and large greeting cards that satisfies the given constraints and generates the highest total profit. Let's assume the art club buys x boxes of small cards and y boxes of large cards. The constraints are:
x + y ≤ 350 (total cards constraint)10x + 15y ≤ 210 (cost constraint)The objective is to maximize profit, given by:
52.20x + 85y
We need to solve this linear programming problem to find the values of x and y that maximize profit. The optimal solution will provide the number of boxes of small and large cards that should be bought to maximize profit while staying within the given constraints.
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Gina's doctor told her the standardized score (z-score) for her systolic blood pressure, as compared to the blood pressure of other women her age, is 1.50. Which of the following is the best interpretation of this standardized score?
a. Gina's systolic blood pressure is 150.
b. Gina's systolic blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age.
c. Gina's systolic blood pressure is 1.50 above the average systolic blood pressure of women her age.
d. Gina's systolic blood pressure is 1.50 times the average systolic blood pressure for women her age.
e. Only 1.5% of women Gina's age have a higher systolic blood pressure than she does.
Answer:
Option b) Gina's systolic blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age.
Step-by-step explanation:
We are given the following in the question:
The distribution of systolic blood pressure of other women is a bell shaped distribution that is a normal distribution.
z-score = 1.50
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
Let x be the Gina's systolic blood pressure.
Thus, we can write:
[tex]1.50 = \displaystyle\frac{x-\mu}{\sigma}\\\\x = 1.5\sigma + \mu \\\text{where }\sigma \text{ is the standard deviation and }\\\mu \text{ is the mean for the given distribution of blood pressure.}[/tex]
Thus, we can write Gina's blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age.
Option b) Gina's systolic blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age.
Gina's z-score of 1.50 indicates that her systolic blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age.
Explanation:The best interpretation of Gina's standardized score (z-score) of 1.50 is option B: Gina's systolic blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age. Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.
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Question 19
1 pts
A boat sails 60 km east, then 10 km south, and then 20 km
west. The distance, in kilometers, of the boat from its
starting point is:
Answer: 10√17
Step-by-step explanation:
The movement of the boat takes the shape of a trapezium as shown in the attached photo.
The distance of the boat from its starting point is represented by x kilometers.
To determine the distance, x, we would apply Pythagoras theorem on the right angle triangle ABC formed. It is expressed as
Hypotenuse² = opposite side² + adjacent side². It becomes
x² = 40² + 10² = 1600 + 100
x² = 1700
x = √1700 = √100 × √17
x = 10√17
Everybody in Ourtown is a fool or a knave or possibly both. 70% of the citizens are foolsand 85% are knaves. One citizen is randomly selected to be mayor. What is the probabilitythat the mayor is both a fool and a knave?
Answer:
[tex] P( F \cap K) =0.7+0.85 -1=0.55[/tex]
Step-by-step explanation:
For this case we can define some notation first:
F ="One person is fool "
K="One person is knave"
And we have the following probabilities given:
[tex] P(F) = 0.7 , P(K) =0.85[/tex]
And from the given condition that everyone is fool or knave we can deduce that:
[tex] P(K UF) =1[/tex]
Solution to the problem
For this case we want to find this probability:
[tex] P( F \cap K)[/tex]
And we can use the total probability rule given by:
[tex] P(K \cup F) = P(F) +P(K) -P(K \cap F)[/tex]
And replacing the values that we have we got:
[tex] 1 = 0.7+0.85 -P(K \cap F)[/tex]
And if we solve for [tex] P( F \cap K)[/tex] we got:
[tex] P( F \cap K) =0.7+0.85 -1=0.55[/tex]
The total cost incurred by time t in the production of a certain commodity is f(t) dollars. The number of products produced by time t is g(t) units. What does the function f(t)/g(t) represent?
Answer:
f(t)/g(t) represents the average cost of producing a unit of commodity between the time frame 0-t.
Step-by-step explanation:
f(t) is cost in dollar while g(t) is in unit. f(t)/g(t) will be cost per unit.
In other words f(t)/g(t) is the total cost spent in time t divided by the amount of commodity produced in units produced in time t.
An amusement park charges admission plus a fee for each ride. Admission plus two rides costs $10. Admission plus five rides cost $16. What is the charge for admission and the cost of a ride?
Answer:the charge for admission is $6 and the cost of a ride is $2
Step-by-step explanation:
Let x represent the charge for admission.
Let y represent the cost of a ride.
An amusement park charges admission plus a fee for each ride. Admission plus two rides costs $10. This means that
x + 2y = 10 - - - - - - - - - - - - - 1
Admission plus five rides cost $16. This means that
x + 5y = 16 - - - - - - - - - - -- - -2
Subtracting equation 2 from equation 1, it becomes
- 3y = - 6
y = - 6/- 3
y = 2
Substituting y = 2 into equation 1, it becomes
x + 2×2 = 10
x = 10 - 4 = 6
If the sum of the interior angles of a regular polygon measures up to 1440 degrees, how many sides does the polygon have?
1. 10 sides
2. 8 sides
3. 12 sides
4. 9 sides
5. None of these
Answer:
Option 1) 10 sides
Step-by-step explanation:
We are given a regular polygon. The sum of interior angles measure upto 1440 degrees.
Since it is a regular polygon, it satisfies the following properties:
All the sides are equal.All interior angle have same measure.Let the regular polygon have n sides.
Then, the sum of interior angle is given by:
[tex](n-2)\times 180^\circ[/tex]
Putting the values, we get,
[tex](n-2)\times 180 = 1440\\\\n-2 = \dfrac{1440}{180}\\\\n-2 = 8\\n = 8 + 2\\n =10[/tex]
Thus, there are 10 sides. The regular polygon is a regular decagon.
Answer:
Decagon
Step-by-step explanation:
Found other sources saying the same thing
Combs and Stratton invested $140,000 and $360,000, respectively, in a partnership. They agreed to share net income in proportion to their investments. If the partnership's net income last year was $43,500, how much was each partners share of the net income?
Answer:
Comb's share will be = $12,180
Stratton's share will be = $31,320
Step-by-step explanation:
Given:
Comb's investment in the partnership = $140,000
Stratton's investment in the partnership = $360,000
The net income is shared in proportions of their investment.
Net income last year = $43,500
To find the share of each partner of the net income.
Solution:
Ratio of the investments of Comb to Stratton = [tex]\frac{140,000}{360,000}[/tex][tex]= \frac{14}{36}=\frac{7}{18}[/tex] (Simplest ratio)
Thus, the investments must be shared in the ratio of 7 : 18
Let Comb's share in dollars be = [tex]7x[/tex]
Then, Stratton's share in dollars = [tex]18x[/tex]
Total net income can be given as = [tex]7x+18x=25x[/tex]
Net income = $43,500
So, we have:
[tex]25x=43,500[/tex]
Dividing both sides by 25.
[tex]\frac{25x}{25}=\frac{43,500}{25}[/tex]
∴ [tex]x=1740[/tex]
So, Comb's share will be = [tex]7\times 1740 = \$12,180[/tex]
Stratton's share will be = [tex]18\times 1740 = \$31,320[/tex]
Working as an insurance salesperson, Ilya earns a base salary and a commission on each new policy, so Ilya’s weekly income, II, depends on the number of new policies, n, he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies, and earned $920. Find an equation for I(n), and interpret the meaning of the components of the equation.
Answer:
l(n) = 80n + 520
Step-by-step explanation:
From the information given in the question, there is a relationship between the number of new policy sold, n, and earning, I
For 3 new policies, he earned $760
For 5 new policies, he earned $920.
The rate of change of IIya's earning with respect to number of new policy sales is
[tex]m = \frac{dI}{dn}[/tex]
[tex]m = \frac{920 - 760}{5 - 3}[/tex]
m = $160 / 2 policies
m = $80 / policy
The linear equation for the relationship is;
l(n) = mn + b
I(n) is Ilya’s weekly income which is a function of the number of new policies, n
m is the rate of change of I with respect to n
n is the number of new policies,
b is the intial function which is IIya's income when n equals zero
Recall, Ilya earns a commission of $80 for each policy sold during the week. (m = $80 per policy)
l(n) = 80n + b
To complete the relationship l, we need to calculate the initial value b.
For 3 new policies, he earned $760,
760 = 80(3) + b
760 = 240 + b
b = 760 - 520
b = 520
The final equation is l(n) = 80n + 520
From the final equation, we can deduce that Ilya’s weekly salary is $520 and he earns an additional $80 commission for each new policy sold.
Final answer:
Ilya's weekly income I(n) can be determined by constructing an equation from the two given points of data, namely I(3) = $760 and I(5) = $920. By solving the system of linear equations, we find that Ilya's base salary is $520 and his commission per policy is $80. The income equation is I(n) = $520 + $80n.
Explanation:
To determine the equation for Ilya's weekly income I(n), we need to establish the relationship between the number of policies sold (n) and the total income (I). Given that Ilya earned $760 for selling 3 policies and $920 for selling 5 policies, we can set up the following two equations based on the formula I(n) = base salary + (commission per policy × n):
1) 760 = base salary + (commission per policy × 3)
2) 920 = base salary + (commission per policy × 5)
To solve this system of equations, we use the method of elimination or substitution. By subtracting the first equation from the second, we can find the commission per policy. Then, we can substitute that value back into either equation to find the base salary. Once we have both values, we can express the equation for Ilya's weekly income as I(n) = base salary + (commission per policy × n).
Step-by-step solution:
Subtract the first equation from the second: 920 - 760 = (base salary + 5× commission) - (base salary + 3× commission)Simplify to find the commission per policy: 160 = 2× commission; hence commission = 80.Substitute the commission value into the first equation: 760 = base salary + 3× 80.Calculate base salary: base salary = 760 - 240 = 520.Formulate the income equation: I(n) = 520 + 80n.Therefore, Ilya's weekly income depends on the base salary of $520 and an additional commission of $80 per new policy sold. The income equation I(n) is both the total of these two components and represents how Ilya's income scales with the number of policies he sells.
A rain barrel can hold 12 gallons of water . Before a storm 2 1/5 gallons of water were in the barrel . The storm added another 6 3/5 gallons of water to the barrel . How many more gallons of water can that barrel hold
Answer:
The rain barrel an hold [tex]3\frac{1}{5}\ gallons[/tex] of water more.
Step-by-step explanation:
Given:
Amount of water barrel can hold = 12 gallons
Amount of water in the barrel before storm = [tex]2\frac{1}{5}\ gallons[/tex]
[tex]2\frac{1}{5}\ gallons[/tex] can be Rewritten as [tex]\frac{11}{5}\ gallons[/tex]
Amount of water in the barrel before storm = [tex]\frac{11}{5}\ gallons[/tex]
Amount of water storm added = [tex]6\frac{3}{5}\ gallons.[/tex]
[tex]6\frac{3}{5}\ gallons.[/tex] can be Rewritten as [tex]\frac{33}{5}\ gallons.[/tex]
Amount of water storm added = [tex]\frac{33}{5}\ gallons.[/tex]
we need to find the amount of water barrel can hold more.
Solution:
Now we can say that;
the amount of water barrel can hold more can be calculated by Subtracting the sum of Amount of water in the barrel before storm and Amount of water storm added from Amount of water barrel can hold.
framing in equation form we get;
the amount of water barrel can hold more = [tex]12-(\frac{11}{5}+\frac{33}{5})= 12-\frac{11+33}{5}= 12- \frac{44}{5}[/tex]
Now we can see that 1 number is whole number and other is fraction.
So we will make the whole number into fraction by multiplying the numerator and denominator with the number in the denominator of the fraction.
so we can say that;
the amount of water barrel can hold more = [tex]\frac{12\times5}{5}-\frac{44}{5} = \frac{60}{5}-\frac{44}{5}[/tex]
Now we can see that denominator is common so we can subtract the numerator.
the amount of water barrel can hold more = [tex]\frac{60-44}{5}=\frac{16}{5}\ gallons \ OR \ \ 3\frac{1}{5}\ gallons[/tex]
Hence The rain barrel an hold [tex]3\frac{1}{5}\ gallons[/tex] of water more.
Final answer:
To find out how many more gallons of water the barrel can hold, subtract the total current water in the barrel from its maximum capacity.
Explanation:
In the question, it is asked how much more water a rain barrel can hold after it has been partially filled. To find this, we need to subtract the amount of water already in the barrel from its total capacity. Initially, the barrel contains 2 1/5 gallons, and the storm adds another 6 3/5 gallons.
We first convert these to improper fractions to make the addition easier.
The rain barrel can hold 12 gallons of water.
Before the storm, there were 2 1/5 gallons in the barrel.
The storm added 6 3/5 gallons of water to the barrel.
To find out how many more gallons of water can the barrel hold, we need to calculate: 12 - (2 1/5 + 6 3/5).
12 - (2 1/5 + 6 3/5) = 12 - (2.2 + 6.6) = 12 - 8.8 = 3.2 gallons.
Each of the possible five outcomes of a random experiment is equally likely. The sample space is {a, b, c, d, e}. All outcomes are equally likely. Let A denote the event {a, b}, and let B denote the event {c, d, e}. Determine the following probabilities:
a) P(A)
b) P(B)
c) P(A')
d) P(A∪B)
e) P(A∩B)
Answer:
a) [tex] p(A) = \frac{2}{5}[/tex]
b) [tex] p(B) =\frac{3}{5}[/tex]
c) [tex] p(A') = 1-p(A) = 1-\frac{2}{5} = \frac{3}{5}[/tex]
d) The probability for intersection on this case is 0 because the sets A and B not have any element in common, so then we have this
[tex] P(AUB) = P(A) +P(B) -0 = \frac{2}{5} +\frac{3}{5} =1[/tex]
e) The intersection for this case is the empty set between the sets A and B so for this reason the probability is 0
P(A∩B)=0
Step-by-step explanation:
For this case we have the following sample space:
[tex] S= [a,b,c,d,e][/tex]
And we have defined the following events:
[tex] A= [a,b][/tex]
[tex] B= [c,d,e][/tex]
For this case we can find the probabilities for each event using the following definition of probability:
[tex] p =\frac{Possible cases}{total cases}[/tex]
The total cases for this case are 5 , the possible cass for A are and for B are 3.
Usign this we have this:
[tex] p(A) = \frac{2}{5}, p(B) = \frac{3}{5}[/tex]
Then we can find the following probabilites:
a) P(A)
[tex] p(A) = \frac{2}{5}[/tex]
b) P(B)
[tex] p(B) =\frac{3}{5}[/tex]
c) P(A')
Using the complement rule we have this:
[tex] p(A') = 1-p(A) = 1-\frac{2}{5} = \frac{3}{5}[/tex]
d) P(A∪B)
For this case we can use the total probability rule and we got:
[tex] P(AUB) = P(A) +P(B) -P(A and B)[/tex]
The probability for intersection on this case is 0 because the sets A and B not have any element in common, so then we have this
[tex] P(AUB) = P(A) +P(B) -0 = \frac{2}{5} +\frac{3}{5} =1[/tex]
e) P(A∩B)
The intersection for this case is the empty set between the sets A and B so for this reason the probability is 0
P(A∩B)=0
The probability of each event in a random experiment is calculated by the ratio of the favorable outcomes to the total outcomes. The answer for each of the given events are: P(A)=2/5, P(B)=3/5, P(A')=3/5, P(A∪B)=1, P(A∩B)=0.
Explanation:In the given random experiment, there are five equally likely outcomes: {a, b, c, d, e}. The event A consists of outcomes {a, b} and the event B consists of outcomes {c, d, e}. The probability of an event can be calculated by the ratio of the number of favorable outcomes to the total number of outcomes.
a) The probability of event A, P(A), is determined by the ratio of the number of outcomes in A to the total outcomes. Since A has 2 outcomes (a and b) and there are 5 total outcomes, the P(A) = 2/5.
b) The probability of event B, P(B), is determined in a similar manner. Since B has 3 outcomes (c, d and e) and there are 5 total outcomes, the P(B) = 3/5.
c) The probability of not A, P(A'), represents all outcomes not in A. Hence, since all outcomes in B and E are not in A, P(A') = P(B) = 3/5.
d) The probability of A or B, P(A∪B), means the probability of either event A or B occurring (or both). Since A and B include all of the outcomes in the sample space, P(A∪B) = 1.
e) The probability of A and B, P(A∩B), is the probability of both event A and event B occurring simultaneously. However, A and B have no common outcomes, so P(A∩B) = 0.
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Find the ratio of the circumference of a circle with radius 1.125 to the circumference of a circle with diameter 2.25. Express your answer in the form a:b where a and b are relatively prime positive integers.
Step-by-step explanation:
Circumference of a circle = πD, where D is the diameter.
Diameter of circle 1 = 1.125
Circumference of circle 1 = π x 1.125
Diameter of circle 1 = 2.25
Circumference of circle 1 = π x 2.25
[tex]\texttt{Ratio of circumferences = }\frac{\pi \times 1.125}{\pi \times 2.25}\\\\\texttt{Ratio of circumferences = }\frac{1}{2}[/tex]
Circumference of circle 1 : Circumference of circle 2 = 1 : 2
Answer:
1:1
Step-by-step explanation:
Use a form of the distributive property to rewrite the algebraic expression without parentheses. one third left parenthesis 4 x minus 15 right parenthesis 1 3(4x−15) one third left parenthesis 4 x minus 15 right parenthesis 1 3(4x−15)equals=nothing (Use integers or fractions for any numbers in the expression. Simplify your answer.)
If a farmer can grow 100 tubs of grapefruit or 250 tubs of oranges per acre of land, what is the opportunity cost of growing one orange?a) 0.4 of an orange b) 2.5 oranges c) 0.4 of a grapefruit d) 2.5 grapefruits
Answer:
Option A. 0.4 of an orange
Step-by-step explanation:
Formula to calculate the opportunity cost is
Opportunity cost = [tex]\frac{\text{Sacrificed}}{\text{Gained}}[/tex]
In this question for the high yield, sacrificed thing is 100 tubs of grapes and gain is to produce 250 tubs oranges.
Opportunity cost = [tex]\frac{100}{250}=0.4[/tex] of an orange
Therefore, Option A. 0.4 of an orange, will be the answer.
Determine which lines, if any, must be parallel. If any lines are parallel, explain your reasoning using if-then statements or the name of the property used.
Answer:
a. a║b
b. c║d
c. AB║CD
d. none
Step-by-step explanation:
a. If distinct lines in a plane are perpendicular to the same line, then they are parallel.
__
b. If same-side interior angles are supplementary where a transversal crosses two lines in the same plane, then those two lines are parallel.
__
c. If alternate interior angles are congruent where a transversal crosses two lines in the same plane, then those two lines are parallel. (Here, the measure of the upper angle at A is 180°-78°-67° = 35°, congruent with the lower angle at C. Those two angles are alternate interior angles with respect to lines AB and CD and transversal AC.)
__
d. The marked angles are unrelated to each other, so define nothing about the relationship between lines a and b, or between lines c and d. However, they do mean that if a║b, then c║d.
Parallelism between lines can be determined by applying geometrical principles or postulates via if-then statements, such as the Corresponding Angles Postulate, Alternate Interior Angles Theorem, or the Converse of the Same-Side Interior Angles Theorem, establishing congruity or supplementarity in the context of lines intersected by a transversal.
Explanation:To determine which lines are parallel, you must look for certain geometrical properties or postulates. If-then statements or direct applications of properties such as the corresponding angles postulate, alternate interior angles theorem, or the converse of the same-side interior angles theorem can be used to identify parallel lines.
If two lines are cut by a transversal and the corresponding angles are equal, then the lines must be parallel (Corresponding Angles Postulate). If the alternate interior angles are congruent when two lines are cut by a transversal, then the two lines are parallel (Alternate Interior Angles Theorem). If two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel (Converse of the Same-Side Interior Angles Theorem). Each of these statements is an application of 'if-then' logic.
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Deangelo needs 100 lb of garden soil to landscape a building. And the company's storage area, he finds two cases holding 24 and 2/3 lb of garden soil each,and a third case holding 19 3/8 lb. How much garden soil does D'Angelo still need in order to do the job?
Question is not proper; Proper question is given below;
D'Angelo needs 100 lb of garden soil to landscape a building. In the company’s storage area, he finds 2 cases holding 24 3/4 lb of garden soil each, and a third case holding 19 3/8 lb. How much gardening soil does D'Angelo still need in order to do the job?
Answer:
D'Angelo required [tex]31 \frac{1}{8}\ lb[/tex] more garden soil to do the job.
Step-by-step explanation:
Given:
Total Amount of garden soil needed to do job = 100 lb
Amount of garden soil in 1st case = [tex]24\frac{3}{4}\ lb[/tex]
[tex]24\frac{3}{4}\ lb[/tex] can be rewritten as [tex]\frac{99}{4}\ lb[/tex]
Amount of garden soil in 1st case = [tex]\frac{99}{4}\ lb[/tex]
Amount of garden soil in 2nd case = [tex]24\frac{3}{4}\ lb[/tex]
[tex]24\frac{3}{4}\ lb[/tex] can be rewritten as [tex]\frac{99}{4}\ lb[/tex]
Amount of garden soil in 2nd case = [tex]\frac{99}{4}\ lb[/tex]
Amount of garden soil in 3rd case = [tex]19\frac{3}{8}\ lb[/tex]
[tex]19\frac{3}{8}\ lb[/tex] can be rewritten as [tex]\frac{155}{8}\ lb[/tex]
Amount of garden soil in 3rd case = [tex]\frac{155}{8}\ lb[/tex]
We need to find Amount of garden soil required more.
Solution:
Now we can say that;
Amount of garden soil required more can be calculated by subtracting sum of Amount of garden soil in 1st case and Amount of garden soil in 2nd case and Amount of garden soil in 3rd case from Total Amount of garden soil needed to do job.
framing in equation form we get;
Amount of garden soil required more = [tex]100-\frac{99}{4}-\frac{99}{4}-\frac{155}{8}[/tex]
To solve the fraction we will make the denominator common using LCM.
Amount of garden soil required more = [tex]\frac{100\times8}{8}-\frac{99\times2}{4\times2}-\frac{99\times2}{4\times2}-\frac{155\times1}{8\times1}= \frac{800}{8}-\frac{198}{8}-\frac{198}{8}-\frac{155}{8}[/tex]
Now denominators are common so we will solve the numerator.
Amount of garden soil required more = [tex]\frac{800-198-198-155}{8}=\frac{249}{8}\ lb \ \ OR \ \ 31 \frac{1}{8}\ lb[/tex]
Hence D'Angelo required [tex]31 \frac{1}{8}\ lb[/tex] more garden soil to do the job.
Solve the inequality.
2(4x - 3) 2-3(3x) + 5x?
Ox20.5
Ox22
0 (-0, 0.5]
O (-, 2]
Answer:
Option 1) x ≥ 0.5
Step-by-step explanation:
The given inequality is : 2(4x - 3) ≥ -3(3x) + 5x
And the options are:
1) x ≥ 0.5
2) x ≥ 2
3) (–∞, 0.5]
4) (–∞, 2]
==============================
So, the solution is as following:
2(4x - 3) ≥ -3(3x) + 5x
8x - 6≥ -9x + 5x
8x + 9x - 5x ≥ 6
12 x ≥ 6
x ≥ 6/12
x ≥ 0.5
The answer is option 1) x ≥ 0.5
• What is the pobability of getting exactly 500 heads out of 1000 coin flips? Approximate it to within 5% of the true value without a calculator.
Answer:Probability of getting exactly 500 heads=0.025
Step-by-step explanation:Probability of getting exactly 500 heads= 1000C500(0.5)^1000=0.025
In the process of loading a ship, a shipping container gets dropped into the water and sinks to the bottom of the harbor. Salvage experts plan to recover the container by attaching a spherical balloon to the container and inflating it with air pumped down from the surface. The dimensions of the container are 5.40 m long, 2.10 m wide, and 3.40 m high. As the crew pumps air into the balloon, its spherical shape increases and when the radius is 1.50 m, the shipping container just begins to rise toward the surface. Determine the mass of the container. You may ignore the weight of the balloon and the air in the balloon. The density of seawater is 1027 kg/m3?
Answer:
Step-by-step explanation:
The value of the total mass will be equal to 13.189x10³ kg.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Calculate weight,
B₁ = ρVg
B₁ = (1027) x (4/3Π(1.3)³ x (9.8)
B₁ = 92.622\ kN
Calculate weight,
B₂ = ( 1027 ) x ( 5 x 2.6 x 2.8) x (9.8)
B₂ = 366.35\ kN
The total mass will be,
B₁ + B₂ = mg
m = ( B₁ + B₂ ) / g
m = ( 36635 + 36635 ) / 9.8
m = 13.189 x 10³ kg
Therefore, the value of the total mass will be equal to 13.189x10³ kg.
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Three cards are dealt from a shuffled standard deck of playing cards. Find the probability that the first card dealt is black, the second is red, and the third is black.
Answer: Probability that the first card dealt is black,the second red and the third black is O.127
Step-by-step explanation: Total number of cards=52
Total number of black cards =26
Total number of red cards=26
Probability of pulling black=26/52
Probability of pulling red=26/51
Probability of pulling a mother black=25/50
Probability of pulling 3 cards =26/52×26/51×25/50
16900/132600
=0.127
The probability that the first card dealt is black, the second is red, and the third is black is 13/102.
Explanation:To find the probability that the first card dealt is black, the second is red, and the third is black, we need to consider the total number of possible outcomes and the number of favorable outcomes. Since we are drawing without replacement, we need to calculate the probabilities for each card.
To find the overall probability, we multiply the probabilities of each event together.
(1/2) * (26/51) * (1/2) = 13/102
Therefore, the probability that the first card dealt is black, the second is red, and the third is black is 13/102.
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Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible.
Answer:
1 is the positive number for which the sum of it and its reciprocal is the smallest.
Step-by-step explanation:
Let x be the positive number.
Then, the sum of number and its reciprocal is given by:
[tex]V(x) = x + \dfrac{1}{x}[/tex]
First, we differentiate V(x) with respect to x, to get,
[tex]\frac{d(V(x))}{dx} = \frac{d(x+\frac{1}{x})}{dx} = 1-\dfrac{1}{x^2}[/tex]
Equating the first derivative to zero, we get,
[tex]\frac{d(V(x))}{dx} = 0\\\\1-\dfrac{1}{x^2}= 0[/tex]
Solving, we get,
[tex]x^2 = 1\\x= \pm 1[/tex]
Since x is a positive number x = 1.
Again differentiation V(x), with respect to x, we get,
[tex]\frac{d^2(V(x))}{dx^2} = \dfrac{2}{x^3}[/tex]
At x = 1
[tex]\frac{d^2(V(x))}{dx^2} > 0[/tex]
Thus, by double derivative test minima occurs for V(x) at x = 1.
Thus, smallest possible sum of a number and its reciprocal is
[tex]V(1) = 1 + \dfrac{1}{1} = 2[/tex]
Thus, 1 is the positive number for which the sum of it and its reciprocal is the smallest.
Consider the following conditional statement:
If two angles form a linear pair, then they are supplementary.
Identify the following statements as the converse, inverse, or contrapositive and determine whether each statement is true or false:
a. If two angles are not supplementary, then they do not form a linear pair.
b. If two angles are supplementary, then they form a linear pair.
c. If two angles do not form a linear pair, then they are supplementary.
Answer:
a. contrapositive because it's the converse and inverse. True.
b. converse because it's the reverse of conditional statement. True.
c. That is false so it's not converse, inverse, or contrapositive.
The given statement is: If two angles form a linear pair, then they are supplementary. The inverse is true, the converse is false, and the contrapositive is true.
Explanation:The given statement is: If two angles form a linear pair, then they are supplementary. Let's analyze the options:
a. If two angles are not supplementary, then they do not form a linear pair. This is the inverse of the given statement. It is true because if two angles do not add up to 180 degrees, they cannot form a linear pair.
b. If two angles are supplementary, then they form a linear pair. This is the converse of the given statement. It is false because two supplementary angles may or may not form a linear pair.
c. If two angles do not form a linear pair, then they are supplementary. This is the contrapositive of the given statement. It is true because if angles do not form a linear pair, that means they do not add up to 180 degrees, and hence, they must be supplementary.
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Evaluate the expression when a=3,b=8, and c=1
B^2-4ac=
Answer:
After evaluating we get [tex]b^2-4ac = 52[/tex].
Step-by-step explanation:
Given:
[tex]b^2-4ac[/tex]
We need to evaluate the expression with a =3, b =8 and c= 1
Solution:
To evaluate the expression we will first substitute the values of a,b and c in the expression we get;
[tex]b^2-4ac = 8^2-4\times3\times1[/tex]
Now by using PEDMAS which states first operation needs to perform here is the exponent function.
so we get;
[tex]b^2-4ac = 64-4\times3\times1[/tex]
Now next operation to be performed is multiplication.
[tex]b^2-4ac = 64-12[/tex]
And finally we will perform subtraction operation.
[tex]b^2-4ac = 52[/tex]
Hence After evaluating we get [tex]b^2-4ac = 52[/tex].
Traveling with the wind, a plane takes 2 1/2 hours to fly a distance of 1500 miles. The return trip of 1500 miles against the same wind speed, takes 3 hours. Find the speed of the plane with no wind and the speed of the wind.
Answer: the speed of the plane with no wind is 500 miles per hour.
the speed of the wind is 100 miles per hour.
Step-by-step explanation:
Let x represent the speed of the plane.
Let y represent the speed of the wind.
Traveling with the wind, a plane takes 2 1/2 = 2.5 hours to fly a distance of 1500 miles. The total speed would be x + y
Distance = speed × time
It means that
1500 = 2.5(x + y)
1500 = 2.5x + 2.5y - - - - - - - - - 1
The return trip of 1500 miles against the same wind speed, takes 3 hours. The total speed is x - y
It means that
1500 = 3(x - y)
1500 = 3x - 3y - - - - - - - - - - - - 2
Multiplying equation 1 by 3 and equation 2 by 2, it becomes
4500 = 7.5x + 7.5y
3000 = 7.5x - 7.5y
Adding both equations, it becomes
7500 = 15x
x = 7500 /15 = 500
Substituting x = 500 into equation 1, it becomes
1500 = 2.5 × 500 + 2.5y
1500 = 1250 + 2.5y
2.5y = 1500 - 1250 = 250
y = 250/2.5 = 100
Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that there is a positive integer that is not the sum of three squares.
The statement 'there is a positive integer that is not the sum of three squares' can be defined mathematically, using predicates, quantifiers, logical connectives, and operators as: ∃ ∈ : ¬(∃,, ∈ : = ² + ² + ²).
Explanation:In order to express the statement that there is a positive integer that is not the sum of three squares, we use predicates, quantifiers, logical connectives, and mathematical operators. Consider the domain of discourse being the set of positive integers. You can express the statement as follows:
∃ ∈ : ¬(∃,, ∈ : = ² + ² + ²)
Overall, this statement corresponds to the claim that there exists some number in the set of positive integers such that no three squares in that set can sum to equal it.
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