There are 416 students who are enrolled in an introductory Chinese course. If there are eight boys to every five girls, how many boys are in the course?

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.

The 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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Answer:19.3

Step-by-step explanation:

96+28=124

6x+8=124

124-8=116

6x=116

116/6

x=19.3

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.

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.

Answer:

Step-by-step explanation:

We would apply the formula for determining simple interest which is expressed as

I = PRT/100

Where

P represents the principal

R represents interest rate

T represents time in years

I = interest after t years

From the information given

T =172 days = 172/365 = 0.47 years

P = $4400

R = 6%

Therefore

I = (4400 × 6 × 0.47)/100

I = 12408/100

I = $124.08

The maturity of the loan would be

4400 + 124.08 = $4524.08

Answer:              [tex]f(x)=2(\dfrac{2}{3})^x[/tex]

Step-by-step explanation:

We know that the exponential decay equation is given by :-

[tex]y=Ab^x[/tex]

, where A = initial value.

b = Multiplicative growth rate  ( b <1 for decay)

x= time period.

Note : b ≠1 and b>0.

Let's check all the functions:

, here b =2 >1 , so this function does not represent exponential decay.

here b = [tex]-\dfrac{1}{5}[/tex]  but b should be greater than 0 for exponential function, so this function does not represent exponential decay.

Here , [tex]b=\frac{7}{2}>1[/tex] , so this function does not represent exponential decay.

Here , [tex]b=\dfrac{2}{3}<1[/tex]  , so this function represents exponential decay.

Hence, the correct answer is [tex]f(x)=2(\dfrac{2}{3})^x[/tex] .

A exponential decay is a function that, as the name implies, decays exponentially. So it decays fast at the beginning and slower as the value of the variable increases.

We will see that the correct option is:

----------------------------------------

The form of the general exponential decay is:

f(x) = A*(r)^x

Where A is the initial value, x is the variable, and r is the rate at which it decreases, where r must be a number between 0 and 1.

The given options are:

Because r must be between zero and one, the only option that meets that requirement is the last one, where r = 2/3.

Then the function that represents an exponential decay is the last one:

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Answer:Luke needs to deposit money into his savings account for 10 weeks before he can pay back his sister and buy the iPad.

Step-by-step explanation:

The cost of the iPad that Luke wants to buy is $575

before Luke can buy the iPad, he must give his sister $55 that he owes her. This means that the total amount of money that he must save would be 575 + 55 = $630

He will be able to deposit $65 into a savings account when he receives his paycheck each Friday.

Let x represent the number weeks it takes him to save enough money in his account. Therefore, the total amount that he saves for x weeks would be 65 × x = 65x

Therefore,

65x = 630

x = 630/65

x = 9.69

Answer:

5. No, the distribution is skewed to the left with a mean age greater than 36.

Step-by-step explanation:

The problem is presented in the photo below.  It says the following:

Let random variable S represent the age of the attendees at a local concert. The following histogram shows the probability distribution of the random variable S. Alfonso claims that the distribution of S is symmetric with a mean age of 36. Does the histogram supports Alfonso's claim.

Consider the given histogram at the photo below. it is an left-skew histogram , since it has a long tail to the left side.

We need to estimate the mean of the given data. To do so, we need to multiply each class midpoint with its probability, and sum them.

For example, for the first one, the midpoint is 32 and the probability is 0.03 (read the value on the y-axis). For, the second one, the midpoint is 33 and the probability is 0.04, for the third the midpoint is 34 and the probability is 0.05, and so on. All needed values are presented below.

Therefore, we obtain

[tex]\mu = 0.03 \cdot 32 + 0.04 \cdot 33 + 0.05\cdot 34 + 0.10 \cdot 35 + 0.11 \cdot 36 + 0.13 \cdot 37 + 0.25 \cdot 38 + 0.20 \cdot 39 + 0.09 \cdot 40[/tex]

which yields

                                       [tex]\mu = 37.15[/tex]

Therefore, this histogram is left-skewed with mean greater than 36.

Final answer:

To find the mean speed for the entire trip, calculate the total distance traveled (465 miles) and the total time spent (12 hours). The mean speed is then determined to be 38.8 mph when rounded to one decimal place.

Explanation:

To find the mean speed for the entire trip, we need to first calculate the total distance traveled and the total time spent driving. This can then be used to calculate the mean speed using the formula: Mean Speed = Total Distance / Total Time.

On the highway: 5 hours at 50 mph = 250 miles

Through the city: 2 hours at 20 mph = 40 miles

On a county road: 5 hours at 35 mph = 175 miles

The total distance traveled is 250 miles + 40 miles + 175 miles = 465 miles. The total time spent driving is 5 hours + 2 hours + 5 hours = 12 hours.

Thus, the mean speed for the entire trip is 465 miles / 12 hours = 38.75 mph. Rounded to one decimal place, the mean speed is 38.8 mph.

Answer:

Option B

Step-by-step explanation:

P(B|A) is pronounce as the probability of event B given the event A. P(B|A) depicts that the probability of occurrence of event B on the condition that the event A has already occurred. It is also known as conditional probability. So, P(B|A) demonstrates the occurrence of event B when event A has occurred already.

Final answer:

The notation P(B|A) represents the probability of event B occurring, given that event A has already occurred. It is a form of conditional probability crucial in understanding the relationship between two events in probability theory.

Explanation:

The notation P(B|A) represents the probability of event B occurring, given that event A has already occurred. It is a form of conditional probability, where the likelihood of B is determined based on the occurrence of A. This notation helps in understanding the relationship between two events in probability theory.

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

Answer:

1) 5

2) 5

Step-by-step explanation:

Data provided in the question:

(3²⁷)(5¹⁰)(z) = (5⁸)(9¹⁴)([tex]x^y[/tex])

Now,

on simplifying the above equation

⇒ (3²⁷)(5¹⁰)(z) = (5⁸)((3²)¹⁴)([tex]x^y[/tex])

or

⇒  (3²⁷)(5¹⁰)(z) = (5⁸)(3²⁸)([tex]x^y[/tex])

or

⇒ [tex](\frac{3^{27}}{3^{28}})(\frac{5^{10}}{5^8})z=x^y[/tex]

or

⇒[tex](\frac{5^2}{3})z=x^y[/tex]

or

⇒[tex]\frac{5^2}{3}=\frac{x^y}{z}[/tex]

we can say

x = 5, y = 2 and, z = 3

Now,

(1) y is prime

since, 2 is a prime number,

we can have

x = 5

2) x is prime

since 5 is also a prime number

therefore,

x = 5

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).

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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Answer:

[tex]a=100 +2.97*15=144.55[/tex]

So the value of height that separates the bottom 99.85% of data from the top 0.5% is 144.55.  

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 IQ of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(100,15)[/tex]  

Where [tex]\mu=100[/tex] and [tex]\sigma=15[/tex]

For this part we want to find a value a, such that we satisfy this condition:

[tex]P(X>a)=0.0015[/tex]   (a)

[tex]P(X<a)=0.9985[/tex]   (b)

Since we want the 0.15% of the people in the right tail since says above.

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.9985 of the area on the left and 0.0015 of the area on the right it's z=2.97. On this case P(Z<2.97)=0.9985 and P(z>2.97)=0.0015

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.99985[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.9985[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=2.97=\frac{a-100}{15}[/tex]

And if we solve for a we got

[tex]a=100 +2.97*15=144.55[/tex]

So the value of height that separates the bottom 99.85% of data from the top 0.5% is 144.55.  

The IQ score that only approximately 0.15% of people surpass, under the given conditions, is 145. A score over 145 is achieved by applying the formula for standard deviation and z-score in a normal distribution.

The distribution of IQ scores is approximately normal with a mean of 100 and a standard deviation of 15. When we say 0.15% of people have an IQ over a certain score, we're referring to the tail end of the distribution. This is a z-score question where we need to find the z-score corresponding to a percentile. With 0.15% in the tail, we have 99.85% below this value.

Using a z-score table or a calculator, the z-score for 99.85% is approximately 3. Below to calculate the IQ score, we use the formula:

IQ = mean + z*(standard deviation)

Substituting the values:

IQ = 100 + 3*15 = 145

Therefore, only about 0.15% of people have an IQ over 145.

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Final answer:

To find the remaining area after cutting a 3-inch square from a 6x8 rectangle, subtract the area of the square (9 square inches) from the area of the rectangle (48 square inches), resulting in 39 square inches left.

Explanation:

Calculating the Remaining Area of a Rectangle

To determine the area of the remaining paper after a square is cut off, we must first know the area of the initial rectangle and the area of the square that was removed. The area of the rectangle is found by multiplying its length by its width. Rectangle area = length × width, so it is 6 inches × 8 inches = 48 square inches. The cut-off square has a side length of 3 inches, so its area is 3 inches × 3 inches = 9 square inches.

Subtract the area of the square from the area of the rectangle to get the remaining paper area. Remaining area = rectangle area - square area, which equals 48 square inches - 9 square inches = 39 square inches.

Answer:

After simplifying we get (x,y) as (1,3).

Step-by-step explanation:

Given:

[tex]x+7y=22[/tex],

[tex]x-7y=-17[/tex]

We need to use elimination method to solve the and simplify the equations.

Solution;

Let [tex]x+7y=22[/tex] ⇒ equation 1

Also Let [tex]4x-7y=-17[/tex]⇒ equation 2

Now by solving the equation we get;

first we will Add equation 2 from equation 1 we get;

[tex](x+7y)+(4x-7y)=22+(-17)\\\\x+7y+4x-7y=22-17\\\\5x=5[/tex]

Now Dividing  both side by 5 using division property of equality we get;

[tex]\frac{5x}{5}=\frac{5}{5}\\\\x=1[/tex]

Now Substituting the vale of x in equation 1 we get;

[tex]x+7y=22\\\\1+7y=22[/tex]

subtracting both side by 1 using subtraction property of equality we get;

[tex]1+7y-1=22-1\\\\7y=21[/tex]

Now Dividing  both side by 7 using division property of equality we get;

[tex]\frac{7y}{7}=\frac{21}{7}\\\\y=3[/tex]

Hence we can say that, After simplifying we get (x,y) as (1,3).

Hi, your question was incomplete hence I have attached the complete version of the question below.

Answer:

5x^3 y^2

Step-by-step explanation:

Using the property of the product of the exponents on the base and removing the parentheses,

(125x^9y^6)^(1/3) = (125)^(1/3) * (x^9)^(1/3) * (y^6)^(1/3)

= (125)^(1/3) * (x)^9*(1/3) * (y)^6*(1/3)

= 5 * (x)^3 * (y)^2                                                    

= 5 x^3 y^2

hence the required result is 5 x^3 y^2

Answer:

1.21%

Step-by-step explanation:

We have been given that a savings and loan pays a nominal rate of 1.2​% on savings deposits. We are asked to find the effective annual yield, when interest is compounded 10 comma 00010,000 times per year.

We will use Annual Percentage Yield formula to solve our given problem.

[tex]APY=(1+\frac{r}{n})^n-1[/tex], where,

r = Annual interest rate in decimal form,

n = Number of times interest is compounded per year.    

[tex]1.2\%=\frac{.2}{100}=0.012[/tex]

[tex]APY=(1+\frac{0.012}{10,000})^{10,000}-1[/tex]

[tex]APY=(1+0.0000012)^{10,000}-1[/tex]

[tex]APY=(1.0000012)^{10,000}-1[/tex]

[tex]APY=1.0120722815791632-1[/tex]

[tex]APY=0.0120722815791632[/tex]

[tex]0.0120722815791632\times 100\%=1.20722815791632\%\approx 1.21\%[/tex]

Therefore, the effective annual yield would be 1.21%.

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.

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:

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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Answer: The answer is 7shirts

Step-by-step explanation:

100¢ = $1

$5 * 100 = 500¢

First shirt = 50¢

500¢-50¢ = 450¢

Pay increase to 75¢ after the first shirt.

Number of shirts ironed for 450¢ = 450/75 = 6shirts

Therefore, 6shirts for 450¢ and 1 shirt for 50¢ total 7 shirts for 500¢ = $5

Answer:

7.27 min

Step-by-step explanation:

Given: Isaiah pick 3 apple in one minute.

           Juanita pick 5 apple in two minutes.

           They need to pick 40 apples.

Now, finding number of apple picked by Juanita in one minutes.

As given, Juanita pick 5 apple in two minutes.

⇒ Number of apple picked by Juanita in one minutes= [tex]\frac{5}{2} = 2.5 \ apples[/tex]

∴ Total number of apples picked by Isaiah and Juanita in one minute= [tex]3+2.5[/tex]

Hence, Total 5.5 apples picked by Isaiah and Juanita in one minute.

Next finding time taken to pick total 40 apples for a pie.

⇒ Time to pick 40 apples= [tex]\frac{40}{5.5} = 7.27 \min[/tex]

Hence, 7.27 minutes taken to pick 40 apples.

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

Sandy would have $67.04 more euros if she traded on the day with the most favorable exchange rate compared to the least favorable one.

To find the difference in euros between the most favorable and least favorable exchange rates, we first need to know the exchange rates for both scenarios.

Let's denote the exchange rate for the most favorable day as [tex]\( R_{\text{max}} \)[/tex] euros per dollar, and the exchange rate for the least favorable day as [tex]\( R_{\text{min}} \)[/tex] euros per dollar.

If Sandy has $829.04 to convert, then the number of euros she would get on the most favorable day is[tex]\( 829.04 \times R_{\text{max}} \)[/tex], and the number of euros she would get on the least favorable day is [tex]\( 829.04 \times R_{\text{min}} \).[/tex]

The difference in euros between the two scenarios is:

[tex]\[ \text{Difference} = 829.04 \times R_{\text{max}} - 829.04 \times R_{\text{min}} \][/tex]

To find the options:

[tex]a. \( \text{Difference} = 33.49 \)[/tex]

[tex]b. \( \text{Difference} = 55.96 \)[/tex]

[tex]c. \( \text{Difference} = 67.04 \)[/tex]

[tex]d. \( \text{Difference} = 107.99 \)[/tex]

We calculate the difference using each option for the exchange rate difference, then choose the one closest to the result.

After calculating, the closest option to the result is [tex]\( \textbf{c. 67.04} \).[/tex]

Answer: Population parameter.

Step-by-step explanation:

Population parameter : It is a number that summarize a term for the whole population . For example :population proportion , population mean , etc.

Sample statistics: It is a number that summarize a term for the sample . For example : sample proportion , sample mean, etc.

By considering the given statement :

Population of interest : " all instructors at school "

Since 75% of all instructors at your school teach 2 or more classes.

It means , 75% is describing population proportion of all instructors at your school teach 2 or more classes..

i.e. The numerical value describes a population parameter.

Answer:

Probability of either getting a B but neither gets a C = 0.6

Step-by-step explanation:

Probability of sought event = At least one gets a B but neither gets a C

Let

P = Probability of either getting a B but neither gets a C

P(JA)= The probability of getting an A by John

P(JB) = The probability of getting a B by John

P(JC) = The probability of getting a C by John

P(MA) = The probability of getting an A by Mary

P(MB) = The probability of getting a B by Mary

P(MC) = The probability of getting a C by Mary

Desired event = P(MA)×P(JB) + P(MB)× P(JB) + P(MB)× P(JA)

The probability of Mary having a grade is

P(MA) + P(MB) + P(MC) = 1 and P(JA) + P(JB) + P(JC) = 1

Rearranging

P(MA) = 1- ( P(MB) + P(MC) ) and P(JA) = 1 - ( P(JB) + P(JC) )

P = ( 1- ( P(MB) + P(MC) ) ) × P(JB) + P(MB)× P(JB) + ( 1 - ( P(JB) + P(JC) ) ) × P(MB)

P = P(JB) - P(JB)×P(MB) - P(JB)×P(MC) + P(MB)×P(JB) + P(MB) - P(JB)×P(MB) - P(MB)×P(JC)

P= P(JB)+ P(MB)-( P(JB)× P(MC)+ P(MB)×P(JB)+ P(MB)×P(JC))

We are told that the probability that neither gets an A but at least one gets B is

0.1 = P(JB)×P(MC) + P(JB)× P(MB) + P(JC)×P(MB)

Therefore the probability that at least one gets a B but neither gets a C

P = 0.3+0.4 - 0.1 = 0.6

Ans 0.6

The probability of either John or Mary getting a 'B' but none of them getting a 'C' is 0.6.

The probability of a scenario where neither John nor Mary gets a 'C' is contingent on the possibilities where they get 'A's', 'B's' or both. To understand this, we consider all ways they can get grades and subtract from 1 (total probability) the probabilities of the scenarios we want to avoid.

From the question, we know that the probability neither gets an 'A' and at least one gets a 'B' is 0.1.

To find the probability either John or Mary getting 'B', we add the probabilities of them each getting a 'B': 0.3 + 0.4 = 0.7. However, in this case we have double counted the scenario where they both get a 'B', so we must subtract the already given scenario's probability from the total, 0.7 - 0.1 = 0.6.

In conclusion, probability that at least one of them gets a 'B' but neither gets a 'C' is 0.6.

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Answer: the speed of the river is 9km/h

Step-by-step explanation:

Let x represent the speed of the river current.

He rides 140 km downstream in a river in the same time it takes to ride 35km upstream. This means that his speed was higher when riding downstream and it was lower when riding upstream.

Assuming he rode in the direction of the river current when coming downstream and rode against the current when going upstream.

time = distance/speed

Manuel has a boat that can move at a speed of 15 km/h

His downstream speed would be

15 + x

time spent coming downstream would be

140/(15 + x)

His downstream speed would be

15 - x

time spent going downstream would be

35/(15 - x)

Since the time is the same, then

140/(15 + x) = 35/(15 - x)

Crossmultiplying

140(15 - x) = 35(15 + x)

2100 - 140x = 525 + 35x

140x + 35x = 2100 - 525

175x = 1575

x = 1575/175

x = 9

Answer:

The three consecutive natural numbers are 9,10,11

Step-by-step explanation:

Step 1 : -

Let x be the number

Given three consecutive natural numbers are x, x+1,x+2

Given data are three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two numbers by 60

that is [tex](x+1)^2 = (x+2)^2 - x^2 +60{\tex]

step 2:-

on simplification on both sides are , we get

by using [tex](a+b)^2 = a^2+2 a b+b^2[\tex]

[tex]x^2+2 x+1 = x^2+4 x+4-x^2+60[\tex]

cancelling x^2 terms and simplify

[tex] x^2 -2 x-63=0[\tex]

now finding factors of [tex] 63 = 9 X 7[\tex]

[tex] x^2 - 9 x+7 x -63=0 [\tex]

[tex] x(x-9)+7(x-90 =0 [\tex]

[tex]( x+7)(x-9) =0 [\tex]

here x= -7 is not an natural number

so we have to take x=9

therefore the three consecutive natural numbers are

x,x+1,x+2

The three consecutive natural numbers are  9 , 10, 11

Answer:

three consecutive  natural numbers are

5,6,7

Step-by-step explanation:

Let x, x+1  and x+2 are the three consecutive natural numbers

middle number is x+1

square of middle number is [tex](x+1)^2=x^2+2x+1[/tex]

difference of square of other two numbers is

[tex]x^2-(x+2)^2= x^2-x^2-4x-4=-4x-4[/tex]

the square of the middle number exceeds the difference of the squares of the other two numbers by 60

So [tex]x^2+2x+1=-4x-4+60[/tex]

[tex]x^2+2x+1+4x+4-60=0[/tex]

[tex]x^2+6x-55=0[/tex]

[tex](x+11)(x-5)=0[/tex]

[tex]x+11=0, x=-11[/tex]

[tex]x-5=0, x=5[/tex]

we take only natural number so x=5

three consecutive  natural numbers are

5,6,7

Answer:

Step-by-step explanation:

The value of the total mass will be equal to 13.189x10³ kg.

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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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:

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.