Patrick bought a new outfit to wear for
school pictures. He bought a pair of pants
for $17.99, a shirt for $12.59, a belt for
$10.80, and a new pair of shoes for
$24.95. How much money did Patrick
spend altogether?

The given function f(x) = -[tex]14x^(^-^\sqrt3)[/tex] is a power function.

A power function is a function of the form f(x) = [tex]kx^m[/tex], where k and m are constants, and x is the independent variable raised to the power of m.

On the other hand, a polynomial function is a function of the form f(x) = [tex]a_nx^n[/tex] + [tex]a_{n-1}x^{n-1}[/tex] + [tex]...[/tex] + [tex]a_1x[/tex] + a_0, where n is a non-negative integer and [tex]a_n[/tex], [tex]a_{n-1}[/tex], [tex]...[/tex], [tex]a_1, a_0[/tex] are constants.

Since the given function cannot be written in the form of a polynomial function, it is not a polynomial function. Therefore, the correct answer is: Power function.

The given function f(x) = -[tex]14x^(^-^\sqrt3)[/tex]is a power function because it can be written in the form f(x) = , where k = -14 and m = -√3.

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The value of B = [0 0 2; 0 3 -1; 2 0 0]. Since AP = PD, we can solve for A by multiplying both sides by [tex]P^{-1}[/tex]:

A =[tex]P^{-1}[/tex] PD = D = [-1 0; 0 2]

How to find A5?

To find [tex]A^{5}[/tex] we can simply raise A to the fifth power:

[tex]A^{5}[/tex] =[tex]D^{5}[/tex] = [[tex]-1^{5}[/tex] 0; 0 [tex]2^{5}[/tex]] = [-1 0; 0 32]

We can simplify the given equation as follows:

AB + I = [tex]A^{2}[/tex] + B

AB - B = [tex]A^{2}[/tex] - I

B(A - I) = [tex]A^{2}[/tex] - I

B = ([tex]A^{2}[/tex]- I)[tex](A - I)^{-1}[/tex]

To find B, we need to first calculate [tex]A^{2}[/tex] and A - I:

[tex]A^{2}[/tex] = [1 0 1; 0 4 0; 1 0 1]

A - I = [0 0 1; 0 1 0; 1 0 0]

Next, we need to calculate the inverse of A - I:

[tex](A - I)^{-1}[/tex] = [-1 0 1; 0 1 0; 1 0 -1]

Finally, we can substitute these values into the expression for B:

B = ([tex]A^{2}[/tex] - I)([tex](A - I)^{-1}[/tex]

B = ([1 0 1; 0 4 -1; 1 0 1] - [1 0 0; 0 1 0; 0 0 1])([-1 0 1; 0 1 0; 1 0 -1])

B = [0 0 2; 0 3 -1; 2 0 0]

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If the observed t-value exceeds the critical t-value, then the difference between the sample means is most likely statistically significant.

In other words, the difference between the sample means is unlikely to have occurred by chance and is likely due to a real difference in the population means. The direction and magnitude of the difference must be further examined using effect size measures and confidence intervals.

Therefore, the most suitable value to be fit in the blank space of the given incomplete statement is, statistically significant.

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Mr. Knapp initially had 61 stamps.

Given that Mr. Knapp had some stamps,

if he gave 7 stamps to each of his children, now he would have 5 stamps left.  

If he gave 8 stamps to each child, he would need 3 more stamps in order for the children to have an equal number of stamps.  

We need to find that how many stamps did Mr. Knapp had.

Let's solve the problem step by step.

Let's assume Mr. Knapp has "x" stamps initially.

According to the first condition, if he gave 7 stamps to each of his children, he would have 5 stamps left.

This can be expressed as:

x - 7c = 5,

where "c" represents the number of children.

According to the second condition, if he gave 8 stamps to each child, he would need 3 more stamps in order for the children to have an equal number of stamps.

This can be expressed as:

x - 8c = -3.

We now have a system of two equations with two variables:

x - 7c = 5,

x - 8c = -3.

Let's use the elimination method to eliminate "x" from the equations:

(x - 7c) - (x - 8c) = 5 - (-3),

x - 7c - x + 8c = 5 + 3,

c = 8.

Now that we know the value of "c" is 8, we can substitute it back into either of the original equations to find the value of "x."

Let's substitute it into the first equation:

x - 7(8) = 5,

x - 56 = 5,

x = 5 + 56,

x = 61.

Therefore, Mr. Knapp initially had 61 stamps.

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Answer:

(a) We can use the given formula to set up an equation:

C = 5 + 1.5m + B

Mae travelled 6 miles and paid 15.50, so we can substitute these values into the formula to get:

15.50 = 5 + 1.5(6) + B

Simplifying and solving for B, we get:

15.50 = 14 + B

B = 1.50

Therefore, Mae paid a booking fee of 1.50.

(b) We can use the given formula to calculate the cost for Noreen's journey:

C = 5 + 1.5m + B

Noreen travelled 10 miles, so we can substitute this value into the formula to get:

C = 5 + 1.5(10) + B

We don't know the value of B, but we can use the information that the cost of Noreen's journey is proportional to the distance she travelled. This means that the ratio of the cost to the distance travelled should be the same as the ratio of the cost of Mae's journey to the distance she travelled:

C / 10 = 15.50 / 6

Solving for C, we get:

C = 10 * (15.50 / 6) = 25.83 (rounded to two decimal places)

Therefore, Noreen paid 25.83 for her journey.

Sorry for bad handwriting

if i was helpful Brainliests my answer ^_^

The series estimate of the sum is between 1.000020993 and 1.000031132.

The given series is S = [tex]1/1^5 + 1/2^5 + 1/3^5[/tex] + ...

We need to compute the 4th partial sum S4.

S4 =[tex]1/1^5 + 1/2^5 + 1/3^5 + 1/4^5[/tex]

Using a calculator, we get:

S4 ≈ 1.000015873

To estimate the error in using S4 as an approximation of the sum of the series, we can use the integral test. Let f(x) = 1/[tex]x^5[/tex].

Then, ∫f(x) dx from 4 to ∞ = ∫(1/[tex]x^5[/tex]) dx from 4 to ∞

= [-1/([tex]4x^4[/tex])] from 4 to ∞

= (1/[tex]4^4[/tex]) - (1/∞)

= 1/256

Since f(x) is positive, decreasing, and continuous for x ≥ 1, the integral test applies and we have:

S - S4 = ∫f(x) dx from 4 to ∞

≥ 1/256

Therefore, the error in using S4 as an approximation of the sum of the series is at least 1/256.

To find a better estimate of the sum, we can use the following inequality:

S_n + ∫f(x) dx from n+1 to ∞ ≤ S ≤ S_n + ∫f(x) dx from n to ∞

where Sn is the nth partial sum of the series.

For n = 4, we have:

S4 + ∫f(x) dx from 5 to ∞ ≤ S ≤ S4 + ∫f(x) dx from 4 to ∞

Substituting the values, we get:

1.000015873 + ∫(1/[tex]x^5[/tex]) dx from 5 to ∞ ≤ S ≤ 1.000015873 + ∫(1/[tex]x^5[/tex]) dx from 4 to ∞

Simplifying the integral, we get:

1.000015873 + [(1/[tex]4^4[/tex]) - (1/5^4)] ≤ S ≤ 1.000015873 + (1/[tex]4^4[/tex])

Solving for the bounds, we get:

1.000015873 + 0.00000512 ≤ S ≤ 1.000015873 + 0.000015259

1.000020993 ≤ S ≤ 1.000031132

Therefore, a better estimate of the sum is between 1.000020993 and 1.000031132.

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The equation that represents line r is:  4x - 3y = 1

To find the equation of a line, we need either two points or a point and the slope of the line. In this case, we have one point (-2,-3) on the line but not the slope. However, we can still find the equation of the line using the point-slope form of the equation:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Since we don't know the slope, let's leave it as m for now:

y - (-3) = m(x - (-2))

Simplifying this equation gives:

y + 3 = m(x + 2)

Now, we need to eliminate the variable m by substituting another point on the line to solve for it. Let's choose the point (1,1) and substitute it into the equation:

1 + 3 = m(1 + 2)

4 = 3m

m = 4/3

Substitute this value of m back into the point-slope form of the equation:

y + 3 = (4/3)(x + 2)

Multiplying both sides by 3 gives:

3y + 9 = 4x + 8

Subtracting 4x from both sides and simplifying:

4x - 3y = 1

Therefore, the equation that represents line r is:  4x - 3y = 1

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This shows that has a remainder of 217 when divided by 343. Therefore, its last two digits in base 343 are 7 and 217.

To approach this problem, we need to use some basic rules of number systems.

When a number is written in base 10 (decimal), its last digit is determined by its remainder when divided by 10. For example, if a number ends with a 7, we know it leaves a remainder of 7 when divided by 10.

Similarly, when a number is written in base n, its last digit is determined by its remainder when divided by n. For example, if a number in base 5 ends with a 3, we know it leaves a remainder of 3 when divided by 5.

Now let's apply this to the given problem. We know that leaves a remainder of 1 when divided by 7 (since its last digit is 1 in base 7). This means that can be expressed as 7k + 1 for some positive integer k.

Similarly, we know that leaves a remainder of 36 when divided by 49 (since its last two digits are 36 in base 49). This means that can be expressed as 49m + 36 for some positive integer m.

To find the last two digits of in base 343, we need to find its remainder when divided by 343. We can start by expressing as a sum of multiples of 7 and 49:

= (7k + 1) + 6(49m + 36)

= 7k + 294m + 217

Now we need to express this sum as a multiple of 343 plus a remainder. We can do this by dividing 294m + 217 by 343:

294m + 217 = 343m + (294m + 217 - 343m)

= 343m + 7m + 217

= 343(m + 1) + 7m - 126

So we have:

= 7k + 343(m + 1) + 7m - 126

= 343(m + 1) + 7(k + m) - 126

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The expression without the absolute value symbol is:

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

|(x - 19) + 11|

This can be written as,

= |x - 19 + 11|

= |x - 8|

Now,

|x - 8| = x - 8, if x > 8

|x - 8| = 0, if x = 8

|x - 8| = 8 - x, if x < 8

Thus,

The expression without the absolute value symbol is:

|x - 8| = x - 8, if x > 8

|x - 8| = 0, if x = 8

|x - 8| = 8 - x, if x < 8

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Step-by-step explanation:

x/5 = 7/13      multiply both sides of the equation by to get

x = 5 * 7/13  =  35/ 13     then use calculator = ~  2.7

Answer:

7/13 = x/5 can be solved by cross-multiplication of the terms.

13*x = 5 *7

13x = 35

x = 35/13 = 2.69 = 2.7

x = (-b + sqrt(b² - 4ac)) / 2a and x = (-b - sqrt(b² - 4ac)) / 2a

What is quadratic equation?

it's a second-degree quadratic equation which is an algebraic equation in x. Ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term, is the quadratic equation in its standard form. A non-zero term (a 0) for the coefficient of x2 is a prerequisite for an equation to be a quadratic equation.

The quadratic formula gives the roots of a quadratic equation in the form of ax² + bx + c = 0. The formula is:

x = (-b ± sqrt(b² - 4ac)) / 2a

So the roots of ax² + bx + c = 0 are:

x = (-b + sqrt(b² - 4ac)) / 2a

and

x = (-b - sqrt(b² - 4ac)) / 2a

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Answer: 22400

Step-by-step explanation:

The probability that a randomly chosen voter from the survey is under 50 years old is approximately 0.75 or 75%.

To compute the probability that a randomly chosen voter from the survey is under 50 years old, we need to first calculate the total number of voters who are under 50 years old. We can do this by adding the number of voters in the 18-28, 29-38, and 39-49 age groups.

Total number of voters under 50 years old = 847 + 804 + 773 = 2424

Next, we need to calculate the total number of voters in the survey, which is given as 3228.

Using these values, we can calculate the probability that a randomly chosen voter from the survey is under 50 years old as:

Probability = Number of voters under 50 years old / Total number of voters in the survey

Probability = 2424 / 3228

Probability ≈ 0.75 or 75%.

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A confidence interval of proportions has the bounds given by the rule presented as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the variables used to calculated these bounds are listed as follows:

The confidence level is of 90%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.9}{2} = 0.95[/tex], so the critical value is z = 1.645.

The margin of error is given as follows:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

The parameters for this problem are given as follows:

[tex]\pi = 0.626, n = 938[/tex]

Hence the margin of error of the interval is of:

M = 1.645 x sqrt(0.626 x 0.374/938)

M = 0.026.

The lower bound of the interval is of:

0.626 - 0.026 = 0.6.

The upper bound of the interval is of:

0.626 + 0.026 = 0.652.

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The sunglasses cost $38.4178 in England

The sunglasses cost $2.9178 less in the USA

We must convert the costs to a common currency in order to compare the costs of the eyewear in the two nations.

Let's first translate the cost from pounds to dollars:

£26.99 * $1.42/£1 = $38.4178

Therefore, the sunglasses cost $38.4178 in England.

We may now evaluate the costs:

$35.50 is the cost in the USA.

After conversion, the cost in England is $38.4178.

The sunglasses are cheaper in the USA by:

$38.4178 - $35.50 = $2.9178

Therefore, the sunglasses cost $2.9178 less in the USA.

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That's correct. When making a random guess on a multiple-choice question on the SAT test, with five answer choices (a, b, c, d, e), there is a 1 in 5 or 20% chance of being correct.

The SAT (Scholastic Assessment Test) is a standardized test used for college admissions in the United States.

The test measures a student's readiness for college-level work and is generally taken by high school juniors and seniors.

The SAT has several sections, including reading, writing and language, math (with calculator), math (without calculator), and an optional essay. Each section of the SAT includes multiple-choice questions, with four or five answer choices for each question.

When making a random guess for an answer to a multiple-choice question on the SAT, you simply select one of the answer choices without any prior knowledge or reasoning about the question.

To make a random guess, you can choose an answer choice that feels lucky, use a systematic method like always choosing "C" or "B," or close your eyes and point to an answer choice.

However, keep in mind that random guessing is not likely to result in a high score, and may even decrease your score due to penalties for incorrect answers.

Therefore, it is generally recommended to only guess on a question when you can eliminate at least one or two answer choices as incorrect.

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The value of log subscript 5 baseline 125 is 3, since 5 raised to the power of 3 is equal to 125:

5^3 = 125

So log subscript 5 baseline 125 = 3.

The limit is finite and equals 37.

To calculate the limit, we need to know the functions f(x) and g(x). Without that information, we cannot provide an exact answer to the question.

However, we can provide some general guidance on how to approach such limits.

One method is to use algebraic manipulation to simplify the expression, and then use known limits or other techniques to evaluate the result.

Another method is to use L'Hopital's rule, which involves taking the derivatives of the numerator and denominator of a fraction separately and then evaluating the limit of the resulting quotient.

For example, let's suppose that f(x) approaches 3 as x approaches 0, and that g(x) approaches -2 as x approaches -1. Then, we can rewrite the limit expression as:

lim x→0 4f(x) -15g(x+1) - 5 - [tex]12x^2[/tex]

= lim x→0 4(3) - 15(-2) - 5 - [tex]12x^2[/tex] (substituting the limits of f(x) and g(x))

= 12 + 30 - 5 - 0 (simplifying)

= 37

Therefore, the limit is finite and equals 37.

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Given tannen's perspective on gender differences in communication,  Men far more than women negotiate for higher starting salaries because salary is a sign of status. So, correct option is C.

Based on Tannen's perspective on gender differences in communication, it is likely that you would find that men negotiate for higher starting salaries far more than women because salary is a sign of status.

Tannen argues that men are socialized to use language as a way to establish and maintain their status in society, and negotiating for a higher salary is one way for men to assert their status.

Women, on the other hand, are socialized to use language as a way to build and maintain relationships, and negotiating for a higher salary may be perceived as aggressive or confrontational, which goes against societal expectations for women's communication style.

Therefore, option (c) is the most likely answer.

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Complete question is:

Given tannen's perspective on gender differences in communication, what would you expect to find regarding salary negotiation and gender

answer choices:

a. Women negotiate for higher starting salaries far more than men because they wish to prove their worth

b. Men and women negotiate for higher starting salaries about equally

c. Men far more than women negotiate for higher starting salaries because salary is a sign of status

d. None of the above is true

(a) Out of a population of 1 million, we can expect 1000 people to have this form of cancer, as the rate of the disease is 0.001 (0.1% of the population) multiplied by 1 million.
(b) Out of a population of 1 million, 5000 people would test positive for this form of cancer, as the false positive rate is 0.005 (0.5% of the population) multiplied by 1 million.
(c) The probability that a person who tested positive actually has this form of cancer is 16.69%.

(a) Out of a population of 1 million, how many people are expected to have this form of cancer?

To calculate this, we'll multiply the population by the rate of cancer occurrence:
1,000,000 (population) × 0.001 (rate) = 1,000 people with cancer.

(b) Out of a population of 1 million, how many people would test positive for this form of cancer?

First, we need to find out how many false positives there would be:
1,000,000 (population) × 0.005 (false positive rate) = 5,000 false positives.

Next, since the test has a perfect detection rate (1) for individuals with cancer, all 1,000 people with cancer would test positive.

Total positive tests = 1,000 (true positives) + 5,000 (false positives) = 6,000 positive tests.

(c) Bayes' theorem. The probability of having the disease given a positive test result can be calculated as follows:

P(A|B) = P(B|A) * P(A) / P(B)

where A is the event of having the disease, B is the event of testing positive, P(A) is the prior probability of having the disease (0.001), P(B|A) is the probability of testing positive given that you have the disease (1), and P(B) is the probability of testing positive, which can be calculated as the sum of the true positive and false positive rates:

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(B) = 1 * 0.001 + 0.005 * 0.999
P(B) = 0.005995

Now we can plug in the values and solve:

P(A|B) = 1 * 0.001 / 0.005995
P(A|B) = 0.1669

So the probability that a person who tested positive actually has this form of cancer is 16.69%.

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The radius of the cone is 7.79 millimeters when the volume of this cone is 1,271.7 cubic millimeters.

The volume of the cone is 1,271.7 cubic millimeters.

Height of cone is 20 millimeters

We have to find the radius of the cone

Volume of cone = πr²h/3

1,271.7 = 3.14×r²×20/3

1,271.7 =20.93r²

Divide both sides by 20.93

1,271.7/20.93 = r²

60.75= r²

r=7.79

Hence, the radius of the cone is 7.79 millimeters when the volume of this cone is 1,271.7 cubic millimeters.

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The standard error of the mean (SEM) can be calculated using the formula:
SEM = √(population variance) / √(sample size)
Given that the population variance is 169 and the sample size is 100, the SEM can be calculated as:
SEM = √(169) / √(100)
SEM = 13 / 10
SEM = 1.3
So, the standard error of the mean is 1.3.

To find the standard error of the mean, we use the formula:
Standard error of the mean = population standard deviation / square root of sample size

Since we're given the population variance (169), we can first find the population standard deviation by taking the square root of the variance:

Population standard deviation = sqrt(169) = 13

Now we can plug in this value along with the sample size (100) to find the standard error of the mean:
Standard error of the mean = 13 / sqrt(100) = 1.30

Therefore, the answer is 1.30.

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The main difference between a quantile-based interval and a highest posterior density interval for a parameter θ is how they are constructed and interpreted.

A quantile-based interval, also known as a credible interval, is constructed by taking the α/2 and 1-α/2 quantiles of the posterior distribution of the parameter θ. It provides an interval that contains the true value of θ with a specified probability (1-α). In this interval, the probability is uniformly distributed, meaning that the parameter θ is equally likely to be found anywhere within the interval.

On the other hand, the highest posterior density interval (HPDI) is the shortest interval that contains a specified amount of posterior probability (1-α). The HPDI is constructed by including the most probable values of θ in the interval until the desired probability is reached. In this interval, the parameter θ is more likely to be found in regions with higher posterior density.

In summary, the difference lies in the way the intervals are constructed: the quantile-based interval focuses on the quantiles of the distribution, while the HPDI focuses on the highest posterior densities.

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Step-by-step explanation:

to find the area you first need to find the area of the square which is

[tex]14 \times 14 = {196m}^{2} [/tex]

then to find the area of the top triangle its

[tex]14 \times 14 \div 2 = {98m}^{2} [/tex]

then to find the area of the bottom triangle its

[tex](38 - 28) \times 14 -( (38 - 28) \times (14 \div 2)) = {70m}^{2} [/tex]

then add all of them together and its 364 m squared.

Answer: sorry i can't read the question

Step-by-step explanation:

The statement that is not true about a square is this: a It has no right angles.

The untrue statement about squares is that they have no right angles. This statement is false because a square actually has 4 right angles. Each of these angles measures 90 degrees.

The four corners of a square contain these angles. Recall that the angle on a straight line equals 180 degrees and we can obtain this by merging two angles of a square. Finally, the wrong statement of all listed is option A.

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Answer:

It has no right angles.

Step-by-step explanation:

What is untrue about squares

The untrue statement about squares is that they have no right angles. This statement is false because a square actually has 4 right angles. Each of these angles measures 90 degrees.

To develop a 95% confidence interval for the mean monthly rent for all studio apartments in College Town, we can use the following formula:

CI = x ± z*(σ/√n)

where:
- CI is the confidence interval
- x is the sample mean ($490.80)
- z* is the z-score for a 95% confidence level (1.96)
- σ is the known population standard deviation ($55)
- n is the sample size (70)

Plugging in the values, we get:

CI = $490.80 ± 1.96*($55/√70)
CI = $490.80 ± $12.15

Therefore, the 95% confidence interval for the mean monthly rent for all studio apartments in College Town is $478.65 to $503.95. We can be 95% confident that the true population mean monthly rent for all studio apartments in College Town falls within this range.
Based on the provided information, we can construct a 95% confidence interval for the mean monthly rent of studio apartments in College Town. Here are the relevant terms and values:

- Sample size (n): 70 apartments
- Sample mean (x): $490.80
- Population standard deviation (σ): $55

To calculate the confidence interval, we will use the formula:

Confidence interval = x ± (z * (σ / √n))

For a 95% confidence interval, the z-value is 1.96. Now, let's plug in the values:

Confidence interval = $490.80 ± (1.96 * ($55 / √70))

Confidence interval = $490.80 ± ($13.61)

So, the 95% confidence interval for the mean monthly rent for all studio apartments in College Town is approximately $477.19 to $504.41. This means that we can be 95% confident that the true mean rent for all studio apartments in this city falls within this range.

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The length of the chord x in the circle is 8.9 units.

The chord of the circle is marked as x. The length of the chord x can be found as follows:

Let's use Pythagoras's theorem to find the other half of the chord.

c²= a² + b²

where

Therefore,

1  /2 x = √6²- 4²

1 / 2 x = √36 - 16

1 / 2 x = √20

cross multiply

x = 2√20

x = 2 × 4.472135955

x = 8.94427191

x = 8.9 units

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True. The solution of a system of linear equations in two variables is the point (x, y) where the two lines intersect. This point satisfies both equations in the system simultaneously.

A system of linear equations in two variables is a set of two equations that can be written in the form:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

where x and y are the variables and a₁, b₁, c₁, a₂, b₂, and c₂ are constants. The goal is to find the values of x and y that satisfy both equations in the system.

Geometrically, each equation in the system represents a line in the two-dimensional plane. The solution of the system is the point (x, y) where the two lines intersect. This point satisfies both equations simultaneously and is the only point that does so.

To find the solution of a system of linear equations, we can use various methods such as substitution, elimination, or matrices. These methods involve manipulating the equations in the system to eliminate one of the variables and solve for the other. Once we have the value of one variable, we can substitute it back into one of the equations to solve for the other variable. The solution of the system is then the ordered pair (x, y) that satisfies both equations.

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A. Arc Length =[tex]∫√(1 + (dy/dx)^2) dx from x = 1 to x = 2.[/tex]

B. Surface Area = [tex]2π ∫ x^3 * √(1 + (3x^2)^2) dx from x = 1 to x = 2.[/tex]

The integrals for the given curve y = x^3 between x = 1 and x = 2.

a) To find the arc length of the curve, we'll use the arc length formula:

Arc Length =[tex]∫√(1 + (dy/dx)^2) dx from x = 1 to x = 2.[/tex]
First, find the derivative dy/dx:
dy/dx = d(x^3)/dx = 3x^2.

Now, substitute into the formula and set up the integral:
Arc Length =[tex]∫√(1 + (3x^2)^2) dx from x = 1 to x = 2.[/tex]
b) To find the surface area when the curve is revolved about the x-axis, we'll use the surface area formula:

Surface Area = [tex]2π ∫ y * √(1 + (dy/dx)^2)[/tex]dx from x = 1 to x = 2.

Substitute y = x^3 and dy/dx = 3x^2 into the formula and set up the integral:
Surface Area = [tex]2π ∫ x^3 * √(1 + (3x^2)^2)[/tex] dx from x = 1 to x = 2.

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