Answer:
Megan has left with $24.
Step-by-step explanation:
Let the total money she has be 'y'.
Given:
Megan spent two out of five of her money on a doll.
So we can say that;
Money spent on doll = [tex]\frac{2}{5}y[/tex]
Also Given:
half of the remainder on a musical box.
Money spent on musical box = [tex](y-\frac{2}{5}y)\frac{1}2[/tex]
Now we will make the denominator common using LCM we get;
Money spent on musical box =[tex](\frac{5y}{5}-\frac{2y}{5})\times\frac{1}2}=(\frac{5y-2y}{5})\times \frac{1}{2}=\frac{3y}{10}[/tex]
Now Given:
she spent eight dollars more on the doll than on a musical box.
[tex]\frac{2y}{5} = \frac{3y}{10}+8[/tex]
Combining like terms we get;
[tex]\frac{2y}{5}-\frac{3y}{10}=8[/tex]
Now we will make the denominator common using LCM we get;
[tex]\frac{2y\times2}{5\times2}-\frac{3y}{10}=8\\\\\frac{4y}{10}-\frac{3y}{10}=8\\\\\frac{4y-3y}{10}=8\\\\y=8\times10\\\\y=\$80[/tex]
Money spent on doll = [tex]\frac{2}{5}y=\frac{2}{5}\times 80 = \$32[/tex]
she spent eight dollars more on the doll than on a musical box.
So we can say that;
Money spent on musical box = Money spent on doll -8 = [tex]32-8 = \$24[/tex]
Now to find the remaining money left we will subtract Money spent on doll and Money spent on musical box from total money she had.
framing in equation form we get;
remaining money left = [tex]80-32-24= \$24[/tex]
Hence Megan has left with $24.
Write a piece wise function that models this function
The answer is
[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
========================================================
Here's how I got that answer:
Start with the piecewise definition for y = |x|.
[tex]g(x) = \begin{cases}x \text{ if }x \ge 0 \\ -x \text{ if }x < 0\end{cases}[/tex]
Everywhere you see an 'x', replace it with x+2
[tex]g(x+2) = \begin{cases}x+2 \text{ if }x+2 \ge 0 \\ -(x+2) \text{ if }x+2 < 0\end{cases}[/tex]
[tex]g(x+2) = \begin{cases}x+2 \text{ if }x \ge -2 \\ -x-2 \text{ if }x < -2\end{cases}[/tex]
Now tack on "-4" at the end of each piece so that we shift the function down 4 units
[tex]g(x+2)-4 = \begin{cases}x+2-4 \text{ if }x \ge -2 \\ -x-2-4 \text{ if }x < -2\end{cases}[/tex]
[tex]g(x+2)-4 = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
Check out the attached images below. In figure 1, I graph y = x-2 and y = -x-6 as separate equations on the same xy coordinate system. Then in figure 2, I combine them to form the familiar V shape you see with any absolute value graph.
A street is drawn by dilating segment FG¯ about center A with a scale factor greater than 0 but less than 1. Is this an enlargement or a reduction?
Answer: This is an reduction.
Step-by-step explanation:
A dilation a king of transformation that creates an similar image (about a center of dilation) of the actual figure by changing its size with the use of a scale factor(k).It either shrinks or stretches the image.If |k| is greater than 1 then the image is an enlargement .If |k| is less than 1 then the image is an reduction.If |k| is equals to 1 then there is no change in size.Given : A street is drawn by dilating segment [tex]\overline{FG}[/tex] about center A with a scale factor greater than 0 but less than 1.
Then by using (2.) , we can say that this is an reduction.
A ball is thrown into the air with an initial upward velocity of 48dt/s. Its height h in feet after t seconds is given by the equation h(t)=-16t^2+48t+4
A: What height will the ball be after 2 seconds?
B: After how many seconds will the ball reach its maximum haight?
C: What is the balls maximum height?
Answer:
Step-by-step explanation:
The equation used to represent the height of the ball, h in feet after t seconds is expressed as
h = -16t^2 + 48t + 4
A) The height of the ball after 2 seconds would be
h = - 16 × 2² + 48 × 2 + 4
h = - 64 + 96 + 4
h = 36 feet
B)The equation is a quadratic equation. The plot of this equation on a graph would give a parabola whose vertex would be equal to the maximum height travelled by the rocket.
The vertex of the parabola is calculated as follows,
Vertex = -b/2a
From the equation,
a = -16
b = 48
Vertex = - - 48/32= 1.5
So the ball will attain maximum height at 1.5 seconds.
C) The maximum height of the ball would be
h = -16 × 1.5² + 48 × 1.5 + 4
h = - 36 + 72 + 4
h = 40 feet
Final answer:
The ball will be at a height of 64 feet after 2 seconds. It will reach its maximum height of 58 feet after 1.5 seconds.
Explanation:
To solve the problem, we need to use the given quadratic equation for the ball's height h(t) = -16t2 + 48t + 4. This equation models the motion of the ball thrown into the air with an initial upward velocity.
A: Height After 2 Seconds
To find the height of the ball after 2 seconds, we substitute t = 2 into the equation:
h(2) = -16(2)2 + 48(2) + 4
h(2) = -16(4) + 96 + 4 = 64 feet
B: Time to Reach Maximum Height
To determine when the ball reaches its maximum height, we need to find the vertex of the parabola, which occurs at t = -b/(2a) where a=-16 and b=48. So t = -48/(2(-16)) = 1.5 seconds.
C: Ball's Maximum Height
To find the ball's maximum height, we substitute t = 1.5 into the height equation:
h(1.5) = -16(1.5)2 + 48(1.5) + 4
h(1.5) = -16(2.25) + 72 + 4 = 58 feet
A rectangle has a perimeter of 50 m and a side length of L.
a. Express the other dimension of the rectangle in terms of L.
Answer: The other dimension can be expressed as
(50 - 2L)/2
Step-by-step explanation: First and foremost, we would let the other dimension be represented by B. Then, the perimeter of a rectangle is measured as L+L+B+B or better put;
Perimeter = 2L + 2B
Where L is the measurement of the longer side and B is the measurement of the shorter side.
In this case the perimeter of the rectangle measures 50m, and this can now be written as
50 = 2L + 2B
Subtract 2L from both sides of the equation
50 - 2L = 2L - 2L + 2B
50 -2L = 2B
Divide both sides of the equation by 2
(50 - 2L)/2 = B
Answer:
25-L
Step-by-step explanation:
Let W represent the other side length. The perimeter (P) of the rectangle is ...
P = 2(W+L)
Solving for W, we get ...
P/2 = W+L
P/2 -L = W
Filling in the given value for P, we find ...
W = 50/2 -L = 25 -L
The other dimension is (25-L) meters.
The rule for the function g says to use x−2 as the input of function f.
What statement describes the transformation between function f and function g.
A. translation down 2 units
B. translation left 2 units
C. translation right 2 units
D. translation up 2 units
Subtracting a value from the input x shifts the graph that number of units to the right.
The answer would be C.
the sum of two numbers is 53 and the difference is 3 . what are the numbers
Answer:
The answer to your question is 25 and 28
Step-by-step explanation:
Number 1 = x
number 2 = y
Conditions
1) x + y = 53 ------------- l
2) x - y = 3 ------------ ll
Solve this system of equations by elimination
x + y = 53
x - y = 3
2x = 56
Solve for x
x = 56/2
x = 28
Substitute x in equation 2
28 - y = 3
- y = -28 + 3
- y = -25
y = 25
Zoey wants to cover her bedroom floor with carpet squares.Each square has an area of 1 square foot.Her bedroom measures 12 feet by 14 feet.How many carpet squares does Zoey need?
Answer: Zoey needs 168 square feet if carpet squares.
Step-by-step explanation:
Zoey wants to cover her bedroom floor with carpet squares. Each square has an area of 1 square foot.
The formula for determining the area of a rectangle is expressed as
Area = length × width
Her bedroom measures 12 feet by 14 feet. Therefore, the area of her bedroom would be
12 × 14 = 168 square feet.
Therefore, the number of carpet squares that Zoey needs would be
168/1 = 168 square feet
Answer:
168 squares
Step-by-step explanation:
Each square is 1 foot on a side, so along the 14-foot wall, Zoey will need 14 squares. Altogether, Zoey will need 12 rows of 14 squares, so 12×14 = 168 squares.
Which equation represents a direct variation?
y = 2х
y=x+4
y= 1/2x
y=3/x
Answer:option 1 and option 2 represents a direct variation.
Step-by-step explanation:
A direct variation is one in which, as one variable increases in value, the other variable increases in value. Also as one variable decreases in value, the other variable decreases in value.
Looking at the options,
1) y = 2х
When x = 2, y = 2×2 = 4
When x = 3, y = 2×3 = 6
Therefore, y = 2x is a direct variation.
2) y=x+4
When x = 2, y = 2 + 4 = 6
When x = 3, y = 3 + 4 = 7
Therefore, y = x + 4 is a direct variation.
3) y= 1/2x
When x = 2, y = 1/2×2 = 1/4 = 0.25
When x = 3, y = 1/2×3 = 1/6 = 0.17
Therefore, y = 1/2x is not a direct variation.
4) y = 3/x
When x = 2, y = 3/2 = 1.5
When x = 3, y = 3/3 = 1
Therefore, y = 3/x is not a direct variation.
Solve for q. [tex]3\left(q+\dfrac 43\right) = 23[/tex]
Final answer:
To solve for q in the equation 3(q + 4/3) = 23, we distribute the 3, subtract 4 from both sides, and then divide by 3 to find that q is approximately 6.33.
Explanation:
To solve for q in the equation 3(q + \dfrac{4}{3}) = 23, we need to apply some basic algebra principles. First, we distribute the 3 into the parentheses.
3q + 3 \times \dfrac{4}{3} = 23
3q + 4 = 23
Now, subtract 4 from both sides to get 3q alone on one side.
3q = 23 - 4
3q = 19
Last, divide both sides by 3 to solve for q.
q = \dfrac{19}{3}
q = 6.333...
Thus, q is approximately equal to 6.33 when rounded to two decimal places.
The value of q is q= 19 / 3.
Let's solve for q in the equation:
3(q+ 3 / 4)=23
We can solve the equation by distributing the terms, adding/subtracting to both sides, dividing both sides by the same factor, and simplifying.
Steps to solve:
1. Distribute the terms:
3q+4=23
2. Add/subtract to both sides:
3q+4−4=23−4
3q=19
3. Divide both sides by the same factor:
3q / 3 = 19 / 3
4. Simplify:
q= 19 / 3
Therefore, the value of q is q= 19 / 3.
A function f is described by f(x)=A*exp(kx)+B, where A, B and k are constants. Given f(0)=1, f(1)=2, and that the horizontal asymptote of f is -4, the value of k is
Answer:
k = ln (6/5)
Step-by-step explanation:
for
f(x)=A*exp(kx)+B
since f(0)=1, f(1)=2
f(0)= A*exp(k*0)+B = A+B = 1
f(1) = A*exp(k*1)+B = A*e^k + B = 2
assuming k>0 , the horizontal asymptote H of f(x) is
H= limit f(x) , when x→ (-∞)
when x→ (-∞) , limit f(x) = limit (A*exp(kx)+B) = A* limit [exp(kx)]+B* limit = A*0 + B = B
since
H= B = (-4)
then
A+B = 1 → A=1-B = 1 -(-4) = 5
then
A*e^k + B = 2
5*e^k + (-4) = 2
k = ln (6/5) ,
then our assumption is right and k = ln (6/5)
The value of k is [tex]k=ln(\frac{6}{5} )[/tex].
Given function is,
[tex]f(x)=Ae^{kx} +B[/tex]
Substitute [tex]f(0)=1,f(1)=2[/tex] in above equation.
We get,
[tex]A+B=1\\\\Ae^{k}+B=2[/tex]
Given that horizontal asymptote of f is -4.
[tex]\lim_{x \to -\infty} Ae^{kx}+B=-4\\ \\ B=-4[/tex]
So, [tex]A=1-B=1-(-4)=5[/tex]
Substitute value of A and B.
[tex]5e^{k}-4=2\\ \\e^{k} =\frac{6}{5}\\ \\k=ln(\frac{6}{5} )[/tex]
Hence, the value of k is [tex]k=ln(\frac{6}{5} )[/tex].
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All of the students at North High School took a benchmark test. When the administration analyzed the students' grades, they found that the grades were normally distributed and that [blank] of the students received grades with z-scores between 0.15 and 0.85.
Answer:
24.2% students received grades with z-scores between 0.15 and 0.85
Step-by-step explanation:
We are given the following in the question:
The grades of a benchmark test for North High School were normally distributed.
WE have to find the percentage of students that received grades with z-scores between 0.15 and 0.85.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(score between 0.15 < z < 0.85)
[tex]P(0.15 \leq z \leq 0.85)\\\\= P(z \leq 0.85) - P(z \leq 0.15)\\\\\text{Calculating the value from standard normal z-table}\\\\= 0.802 - 0.560 = 0.242 = 24.2\%[/tex]
24.2% students received grades with z-scores between 0.15 and 0.85
When a scatter chart of data shows a nonlinear relationship, the nonlinear model can be expressed as:______.
A) Y = β0 + β1X + (β2X)2 + ε
B) Y = β0 + β1X + β2X2 + ε
C) Y = β0 + β1X + β2X
D) Y = β0 + β1X2 + β2X2 + ε
Answer:
A
Step-by-step explanation:
The linear model can be assessed by the checking the independent variables having power 1 which shows the linear relationship between x and y. For example, as in the option B, C and D, the power of Xi's is one. Whereas the non linear model has the power for independent variables greater than 1. For example, as in option A the model is a quadratic model because X associated with β2 has a power of a 2.
Thus the nonlinear model can be expressed as
Y = β0 + β1X + (β2X)2 + ε.
What is the average miles per gallon (mpg) for all new cars? Using Consumer Reports, a random sample of 35 new cars gave an average of 21.1 mpg.
(a) Identify the variable.
(b) Is the variable quantitative or qualitative?
(c) What is the implied population?
The variable is the average miles per gallon (mpg), which is a quantitative measure. The implied population is all new cars.
Explanation:(a) The variable in this situation is the average miles per gallon (mpg) for all new cars.
(b) The variable is quantitative, as it deals with a numerical measure, i.e., the number of miles a car can travel per a gallon of fuel.
(c) The implied population would be all new cars in general - though it's specified that this is based on a sample from Consumer Reports, which may not cover every single new car in existence.
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The cost of four evening movie tickets is $33.40 the cost of 6 daytime tickets is 39.30 what is the difference between the cost of one evening ticket in one day time ticket
Answer:the difference between the cost of one evening ticket and one day time ticket s $1.8
Step-by-step explanation:
The cost of four evening movie tickets is $33.40. This means that the cost of one evening ticket would be
33.4/4 = $8.35
The cost of 6 daytime tickets is 39.30. This means that the cost of one daytime ticket would be
39.30/6 = $6.55
Therefore, the difference between the cost of one evening ticket and one day time ticket would be
8.35 - 6.55 = $1.8
What is the order of the numbers from least to greatest? A = 4.6 x 10–4 B = 2.4 x 10–3 C = 3.5 x 105 D = 6.3 x 10–4 A. C < A < B < D B. D < A < C < B C. B < C < A < D D. A < D < B < C
Answer:
D = A < D < B < C
Step-by-step explanation:
A = 4.6 x 10-4
Can be written as,
= 0.00046
B = 2.4 x 10-3
Can be written in this form,
= 0.0024
C = 3.5 x 105
Is written as,
350000
D = 6.3 x 10-4
Is also written as,
= 0.00063
A and D are in 4 decimal places and therefore from the 3th decimal place 46 is less than 63 so therefore 0.00046 is less than 0.00063.
B is greater than A and D because B which is 0.0024 is in 3 decimal places. C which is 35000 is the greatest because there are no decimal places and it is in tenth thousand.
So therefore,
A < D < B < C
What is the recursive rule for the sequence 1, −6, 36, −216, ... ? an=6⋅an−1 , a1=1 an=−6⋅an−1 , a1=1 an=−16⋅an−1 , a1=1 an=16⋅an−1 , a1=1
Answer:
Option 2) [tex]a_n = -6(a_{n-1})[/tex]
Step-by-step explanation:
We are given the following sequence in the question:
[tex]1, -6, 36, -216, ...[/tex]
We have to find the recursive relation for the sequence.
[tex]a_1 =1\\a_2 = -6 = -6(1) = -6(a_1)\\a_3 = 36 = -6(-6) = -6(a_2)\\a_4 = -216 = -6(36) = -6(a_3)[/tex]
Thus, continuing in the following manner, we get,
[tex]a_n = -6(a_{n-1})[/tex]
Thus, the recursive rule is given by
Option 2) [tex]a_n = -6(a_{n-1})[/tex]
Answer:
Step-by-step explanation:
Recent research suggests that depression significantly increases the risk of developing dementia later inlife (BBC News, July 6, 2010). In a study involving 949 elderly persons, it was reported that 22% of thosewho had depression went on to develop dementia, compared to only 17% of those who did not havedepression.a. Choose the relevant population and the sample. (You may select more than one answer.) 1. The sample consists of 949 elderly people.2. The population is all elderly people.3. The population is all younger people.4. The sample consists of 949 younger peopleb. Do the numbers 22% and 17% represent the population parameters or sample statistics?
Answer:
a.
Option 1
Option 2
b. Sample statistics
Step-by-step explanation:
a.
The set that includes the list of all possible individual in the interested area of study is termed as population while the portion or a part of population is termed as sample. The given scenario indicates that population is all elderly people from which 949 elderly people are selected for study and so, 949 elderly people are included in sample. So, option 1 and option 2 are correct for indicated scenario.
b.
The percentages 22% and 17% are calculated from 949 elderly people that are indicated as sample. Hence, the percentages 22% and 17% are the measure of sample and so, they represents sample statistics.
Final answer:
The relevant population is all elderly people, and the sample consists of 949 elderly people. The numbers 22% and 17% represent sample statistics.
Explanation:
a. The relevant population is all elderly people, so options 2 and 3 are correct. The sample consists of 949 elderly people, so option 1 is also correct.
b. The numbers 22% and 17% represent sample statistics. The sample statistics are calculated from the data collected in the sample, which in this case is the proportion of elderly persons with depression who went on to develop dementia.
Solve the complex expression and show work if you can
Answer: 2+i
================================
Work Shown:
-3 + 6i - (-5 - 3i) - 8i
-3 + 6i + 5 + 3i - 8i
(-3+5) + (6i+3i-8i)
2+1i
2+i
The simplified expression is in the form a+bi with a = 2, b = 1.
Answer:
2 + i.
Step-by-step explanation:
-3 + 6i - (-5 - 3i) - 8i Distribute the negative over the parentheses:
= -3 + 6i + 5 + 3i - 8i
= - 3 + 5 + 6i + 3i - 8i Now simplify like terms:
= 2 + i.
Is the following variable categorical or quantitative? Collect data from a sample of teenagers with a question that asks ‘‘Do you eat at least five servings a day of fruits and vegetables?""
The survey question 'Do you eat at least five servings a day of fruits and vegetables?' is designed to collect categorical data, as it classifies respondents into groups based on their affirmative or negative answer, rather than providing a numerical value.
Explanation:The question "Do you eat at least five servings a day of fruits and vegetables?" is designed to collect categorical data. This is because the answers to the question will classify respondents into different categories, specifically those who do eat at least five servings of fruits and vegetables per day and those who do not. As such, the data obtained will be qualitative in nature, allowing us to compare and organize individuals based on their dietary habits.
For a more comprehensive understanding, let's compare data types. In contrast to categorical data, a quantitative variable is numeric and can be measured or counted. It can further be subdivided into discrete or continuous data. Quantitative discrete data involve counts of items or occurrences (e.g., the number of classes you take per school year), while quantitative continuous data involve measurements that can take on any value within a given range (e.g., the weights of soups measured in ounces).
Returning to the student's survey question about fruit and vegetable consumption, it is evident that the data collected does not involve counting or measuring numerical values, but rather involves placing respondents into categories based on their dietary habits. Therefore, the variable in question is indeed categorical.
In each case below, a relation on the set {1, 2, 3} is given. Of the three properties, reflexivity, symmetry, and transitivity, determine which ones the relation has. Give reasons.
Answer:
a. is symmetric but not reflexive and transitive
b. is reflexive and transitive but not symmetric
c. is reflexive, symmetric and transitive
Step-by-step explanation:
The cases are missing in the question.
Let the cases be as follows:
a. R = {(1, 3), (3, 1), (2, 2)}
b. R = {(1, 1), (2, 2), (3, 3), (1, 2)}
c. R = ∅
R is defined on the set {1, 2, 3}
R is reflexive if for all x in {1, 2, 3} xRxR is symmetric if for all x,y in {1, 2, 3} if xRy then yRxR is transitive if for all x,y,z in {1, 2, 3} if xRy and yRz then xRza. R = {(1, 3), (3, 1), (2, 2)} is
not reflexive since for x=1, (1,1) is not in Rsymmetric since for all x,y in {1, 2, 3} if xRy then yRxnot transitive because (1, 3), (3, 1) is in R but (1,1) is not.b. R = {(1, 1), (2, 2), (3, 3), (1, 2)} is
is reflexive because (1, 1), (2, 2), (3, 3) is in R
is not symmetric because for (1,2) (2,1) is not in R
is transitive becaue for (1,1) and (1,2) we have (1,2) in R
c. R = ∅ is
reflexive, symmetric and transitive because it satisfies the definitions since there is no counter example.
Two numbers are 10 units away in different directions from their midpoint, m, on a number line. The product of the numbers is -99. Which equation can be used to find m, the midpoint of the two numbers?
Answer:
[tex]m = \dfrac{a^2-99}{2a}[/tex]
Step-by-step explanation:
on a number line, m is the point that is the midpoint of two other points.
the distance between each of the points to the midpoint is 10 units..
if a is the point 10 units less than m
and b is the point 10 units greater than m,
then,
[tex]m = a+10[/tex]
[tex]m = b-10[/tex]
we can add the two equations to form the midpoint formula.
[tex]2m = a+b[/tex]
we also know that the product of both numbers equal -99.
[tex]ab = -99[/tex]
we can substitute either 'a' or 'b' to the equation of m.
[tex]2m = a-\dfrac{99}{a}[/tex]
[tex]m = \dfrac{a^2-99}{2a}[/tex]
and this is the equation for the midpoint of the two numbers.
Answer:
c
Step-by-step explanation:
The parent teacher organization is selling baskets of cookies for a school fundraiser and materials needed to make each basket cost 375 and the baskets are being sold for $10 each if they spent $75 to advertise their site how many baskets must be sold in order to break even
Answer:12 baskets must be sold in order to break even
Step-by-step explanation:
The materials needed by the fundraiser team to make each basket cost $3.75.
Let x represent the number of baskets that the team made and also sold. if they spent $75 to advertise their site, then the total cost for x baskets would be
3.75x + 75
The baskets are being sold for $10 each. It means that the total revenue would be
10x
In order to break even, the revenue must be equal to total cost. Therefore,
10x = 3.75x + 75
10x - 3.75x = 75
6.25x = 75
x = 75/6/25 = 12
What is the answer to 3 1/4 cans of red paint and 3 2/12 cans of yellow paint add up to how many cans of orange paint? I know it is 6 cans of orange paint, but don't know the fraction.
The number of orange cans of paint is 6
Solution:
Given that,
[tex]\text{Number of cans of red paint } = 3\frac{1}{4}\\\\\text{Number of cans of yellow paint } = 3\frac{2}{12}[/tex]
Let us convert the mixed fractions to improper fractions
Multiply the whole number part by the fraction's denominator.
Add that to the numerator.
Then write the result on top of the denominator
[tex]\rightarrow 3\frac{1}{4} = \frac{4 \times 3 + 1}{4} = \frac{13}{4}\\\\\rightarrow 3\frac{2}{12} = \frac{12 \times 3 + 2}{12} = \frac{38}{12}[/tex]
Now we have to add red cans of paint and yellow cans of paint to get orange cans of paint
[tex]\text{Number of cans of orange paint } = \frac{13}{4} + \frac{38}{12}[/tex]
Take L.C.M for denominators
The prime factors of 4 = 2 x 2
The prime factors of 12 = 2 x 2 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 3
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 3 = 12
[tex]\text{Number of cans of orange paint } = \frac{13}{4} + \frac{38}{12}[/tex]
[tex]\text{Number of cans of orange paint } = \frac{13 \times 3}{4 \times 3} + \frac{38}{12}\\\\\text{Number of cans of orange paint } = \frac{39}{12} + \frac{38}{12}\\\\\text{Number of cans of orange paint } = \frac{77}{12} = 6.4 \approx 6[/tex]
Thus the number of orange cans of paint is 6
Three roots of the polynomial equation X^4-4X^3-2X^2 +12 X +9=0 are 3, -1 and -1. Explain why the fourth root must be a real number. Find the fourth root
Answer:
The fourth root is 3
If the 4th root is not a real therefore it must be a complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4 roots.
Therefore the fourth root is real.
The roots are -1 with multiplicity 2 and 3 with multiplicity 2
Therefore it has four roots
Step-by-step explanation:
Given polynomial equation is [tex]X^4-4X^3-2X^2+12X+9=0[/tex]
And also given that 3,-1 and -1 are the roots of the given polynomial equation
To find the fourth root of the polynomial equation and to solve the fourth root is real :
By synthetic division
_3| 1 -4 -2 12 9
0 3 -3 -15 -9
___________________
_-1| 1 -1 -5 -3 0
0 -1 2 3
___________________
1 -2 -3 0
Therefore x-3 and x+1 is a factor
Therefore 3 and -1 are roots
Now we have the quadratic equation [tex]x^2-2x-3=0[/tex]
[tex](x+1)(x-3)=0[/tex]
Therefore x=-1,3 are the roots
Therefore the fourth root is 3
If the 4th root is not a real therefore it must be a complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4 roots.
Therefore the fourth root is real.
The roots are -1 with multiplicity 2 and 3 with multiplicity 2
Therefore it has four roots.
Final answer:
The fourth root of the polynomial equation X⁴-4X³-2X²+12X+9=0 must be real because a polynomial of degree n has n roots, and since we already have real roots, the remaining root must also be real to have a pair. Upon analyzing, the fourth root is found to be 3.
Explanation:
The student has provided three roots of the fourth-degree polynomial equation X⁴-4X³-2X²+12X+9=0: 3, -1, and -1 (the latter being a repeated root). To determine why the fourth root must also be a real number, we can invoke the fundamental theorem of algebra, which states that a polynomial of degree n will have exactly n roots in the complex number system (including real and complex roots). Given that a polynomial with real coefficients will have complex roots that come in conjugate pairs, and since the known roots are all real, the unknown fourth root must also be real to satisfy the theorem.
Let's find the fourth root. The polynomial can be factored using the known roots:
(X-3) - Factor for root 3(X+1)² - Factor for the repeated root -1Therefore, we have the equation (X-3)(X+1)²(X-a)=0, where 'a' is the unknown root. The product of the roots taken one at a time equals the constant term (9) of the polynomial with an alternate sign. This gives us the equation: 3 × -1 × -1 × a = 9. Solving for 'a' yields a=3, which is the fourth root.
Punction gives the distance of a dog
from a post, in feet, as a function of time,
in seconds, since its owner left.
Find the value of $(20) and of f(140).
distance from post in feet
20
40
60
80
100 120 140
Function C gives the cost, in dollars, of buying n apples. What does each expression
or equation represent in this situation?
a. 8) = 4.50
b. C(2)
The distance of a dog from a post and the cost of buying apples can be represented by functions. We can find the values of these functions by substituting specific values and calculating the corresponding outputs.
Explanation:In this question, we are given a function that represents the distance of a dog from a post as a function of time. To find the value of f(20), we substitute 20 into the function and calculate the corresponding distance. To find the value of f(140), we do the same thing, substituting 140 into the function.
f(20) = 20 feet
f(140) = 140 feet
In the second part of the question, we are given a function C that represents the cost of buying n apples. To find the meaning of each expression or equation, we substitute the given value of n and calculate the corresponding cost.
C(8) = $4.50
C(2) represents the cost of buying 2 apples.
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A local school held a charity coat drive for two months the school collected 269 coats in the first month 542 coats were collected in all how many coats did the school collect in the second month
Answer:
School collected 273 coats in second month.
Step-by-step explanation:
Given
Total number of coats collected in 2 months = 542
Number of coats collected in first month = 269
We need to find the number of coat school collected in second month.
Solution:
Now we can say that;
Total number of coats collected in 2 months is sum of Number of coats collected in first month and Number of coats collected in Second month.
Also We can say that;
Number of coats collected in Second month is equal to Total number of coats collected in 2 months minus Number of coats collected in first month.
framing in equation form we get;
Number of coats collected in Second month = [tex]542-269 =273[/tex]
Hence School collected 273 coats in second month.
Solve the system of linear equations. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, and x3 in terms of the parameter t.)
2x1 + x2 − 2x3 = 4
4x1 + 2x3 = 10
−4x1 + 5x2 − 17x3 = −15
Using the process of Gaussian elimination, the system of linear equations is rewritten in the form of a matrix. It is then transformed into the Row-Echelon form, which helps determine possible solutions. The solution for this particular system of equations is x1 = 2, x2 = 2, and x3 = 1.
Explanation:To solve this system of linear equations, you can use a process called
Gaussian elimination
. You start by rewriting the system in augmented matrix. Thus, the system
2x1 + x2 − 2x3 = 4
4x1 + 2x3 = 10
−4x1 + 5x2 − 17x3 = −15
becomes the matrix
[2 1 -2 4]
[4 0 2 10]
[-4 5 -17 -15]
The next step is to convert this matrix into the Row-Echelon form. Once you have a matrix in Row-Echelon form, you can easily see if there are any solutions by looking at the location of the zeros. If there is a row with all zeros on the left and non-zero terms on the right, then there is no solution. If there are infinite many solutions, its row will end with zeros. In this case, the solution is x1 = 2, x2 = 2, and x3 = 1.
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The area of a rectangular plot is 36 square meters. The length of the plot (in meters) is one more than twice its width. Find the length and width of the plot.
length (m) ______.
width (m) ______.
Answer:
4m width and 9m length
Step-by-step explanation:
Let the width of the rectangle be x
Length is 1 more than twice width= 1 + 2x
Area of rectangle is L * B
x(2x + 1) = 36
2x^2 + x = 36
2x^2 + x -36 = 0
2x^2 + 9x - 8x -36 = 0
Solving this:
(2x+9)(x - 4) = 0
X = 4 or -4.5
Distance cannot be negative, so x = 4m
The length is thus 2(4) + 1 = 9m
Final answer:
The width of the plot is 4 meters and the length is 9 meters.
Explanation:
To solve this problem, we can let the width of the plot be x meters. According to the problem, the length of the plot is one more than twice its width, so the length would be 2x + 1 meters. The area of a rectangle is given by the formula A = length * width. So we have the equation (2x + 1) * x = 36. Expanding and rearranging, we get 2x² + x - 36 = 0.
Factoring this quadratic equation, we get (2x + 9)(x - 4) = 0. Setting each factor equal to zero and solving for x, we find x = -4/2 and x = 4. Since the width cannot be negative, we discard x = -4/2 and conclude that the width of the plot is 4 meters. Substituting this value back into the equation for the length, we find the length is 2(4) + 1 = 9 meters.
A ball is thrown into the air from the top of a building. The height, h(t), of the ball above the ground t seconds after it is thrown can be modeled by h(t) = -16t2 + 64t + 80. How many seconds after being thrown will the ball hit the ground?
Answer: 5 seconds
Step-by-step explanation:
Given : A ball is thrown into the air from the top of a building.
The height, h(t), of the ball above the ground t seconds after it is thrown can be modeled by [tex]h(t) = -16t^2 + 64t + 80[/tex] .
When ball reaches the ground , then its height from ground become zero.
i.e. [tex]h(t) = -16t^2 + 64t + 80=0[/tex]
Divide equation by 16 , we get
[tex]-t^2 + 4t + 5=0\\\\\Rightarrow\ t^2-4t-5=0\\\\\Rightarrow\ t^2+t-5t-5=0\\\\\Rightarrow\ t(t+1)-5(t+1)=0\\\\\RIghtarrow\ (t+1)(t-5)=0 \\\\\Rightarrow\ t= -1 , 5[/tex]
Since time cannot be negative , therefore t= 5
Hence, the ball will take 5 seconds ( after being thrown) to hit the ground.
The ball will hit the ground 5 seconds after being thrown.
Settingh(t) equal to zero gives us the equation:
[tex]\[ -16t^2 + 64t + 80 = 0 \][/tex]
To solve this quadratic equation, we can factor out the common factor of -16:
[tex]\[ -16(t^2 - 4t - 5) = 0 \][/tex]
Now we need to factor the quadratic expression inside the parentheses.
We are looking for two numbers that multiply to -5 and add up to -4. These numbers are -5 and +1. So we can write:
[tex]\[ -16(t - 5)(t + 1) = 0 \][/tex]
Setting each factor equal to zero gives us two possible solutions for t
[tex]\[ t - 5 = 0 \quad \text{or} \quad t + 1 = 0 \][/tex]
Solving these, we get:
[tex]\[ t = 5 \quad \text{or} \quad t = -1 \][/tex]
Since time cannot be negative, we discard the solution t = -1
Therefore, the ball will hit the ground 5 seconds after being thrown."
The distribution of the number of people in line at a grocery store has a mean of 3 and a variance of 9. A sample of the numbers of people in line in 50 stores is taken.
(a) Calculate the probability that the sample mean is more than 4? Round values to four decimal places.
(b) Calculate the probability the sample mean is less than 2.5. Round answers to four decimal places.
(c) Calculate the probability that the the sample mean differs from the population mean by less than 0.5. Round answers to four decimal places.
Answer:
a) [tex]P(\bar X >4)=P(Z>\frac{4-3}{\frac{3}{\sqrt{50}}}=2.357)[/tex]
[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]
b) [tex]P(\bar X <2.5)=P(Z>\frac{2.5-3}{\frac{3}{\sqrt{50}}}=-1.179)[/tex]
[tex]P(Z<-1.179)=0.1192[/tex]
c)
[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]
[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/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".
Part a
Let X the random variable that represent the number of people of a population, and for this case we know that:
Where [tex]\mu=3[/tex] and [tex]\sigma=\sqrt{9}=3[/tex]
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
And we want to find this probability:
[tex] P(\bar X >4)= P(z> \frac{4-3}{\frac{3}{\sqrt{50}}})[/tex]
And using a calculator, excel or the normal standard table we have that:
[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]
Part b
[tex] P(\bar X <2.5)= P(z> \frac{2.5-3}{\frac{3}{\sqrt{50}}})[/tex]
And using a calculator, excel or the normal standard table we have that:
[tex]P(Z<-1.179)=0.1192[/tex]
Part c
For this case we want this probability:
[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]
And using a calculator, excel or the normal standard table we have that:
[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]