Brewsky's is a chain of micro-breweries. Managers are interested in the costs of the stores and believe that the costs can be explained in large part by the number of customers patron¬izing the stores. Monthly data regarding customer visits and costs for the preceding year for one of the stores have been entered into the regression analysis and the analysis is as follows:Average monthly customer-visits 1,462Average monthly total costs $ 4,629Regression Results Intercept $ 1,496b coefficient $ 2.08R2 0.868141. In a regression equation expressed as y = a + bx, how is the letter b best described? (CMA adapted)a. The proximity of the data points to the regression line.b. The estimate of the cost for an additional customer visit.c.The fixed costs per customer-visit.d.An estimate of the probability of return customers.2. How is the letter x in the regression equation best described? (CMA adapted)a. The observed customer visits for a given month.b. Fixed costs per each customer-visit.c. The observed store costs for a given month.d. The estimate of the number of new customer visits for the month3. What is the percent of the total variance that can be explained by the regression equation? (CMA adapted)a. 86.8%b. 71.9%c. 31.6%d. 97.7%

The probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph is approximately 0.99997

A. To find the probability that the speed X of a randomly selected vehicle is between 35 and 40 mph, we need to standardize the values and use the standard normal distribution table.

We can standardize the values as follows:

[tex]z1 = \frac{(35 - 36.6)}{1.7} = -0.94[/tex]

[tex]z2 = \frac{(40 - 36.6)}{1.7} = 2.00[/tex]

Using the standard normal distribution table, we find the probability that a standard normal variable is between -0.94 and 2.00 to be approximately 0.7794.

B. To find the probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph, we need to use the central limit theorem.

The central limit theorem tells us that the distribution of sample means will be approximately normal, with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Thus, the mean of the sampling distribution of the sample means is:

u-X=u=36.6

And the standard deviation of the sampling distribution of the sample means is:

[tex]\frac{ σ}{\sqrt{n} } = \frac{1.7}{\sqrt{20} } = 0.3808[/tex]

We can standardize the values using the formula:

[tex]z1 =\frac{35-36.6}{0.3808} = -4.21[/tex]

[tex]z2 =\frac{40-36.6}{0.3808} = 8.92[/tex]

we can find the probability that the standard normal variable is between -4.21 and 8.92 by finding the probability that it is greater than -8.92 (which is essentially 0) and subtracting the probability that it is greater than 4.21 from 1:

P(-4.21 < Z < 8.92) = 1 - P(Z > 4.21) = 1 - 0.00003 = 0.99997

Therefore, the probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph is approximately 0.99997.

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The linear constraints with continuous and integer variables for the following is 100y + x2 ≤ 200.

The linear constraints for the problem are:
x ≥ 100 → y ≤ 100
x ≥ 0
y ≥ 0

y * 100 ≥ x1
(1 - y) * 100 ≥ x1
100y + x2 ≤ 200.

Let x be the investment amount on project 1 (continuous variable) and y be the investment amount on project 2 (continuous variable).

To write the linear constraints for the problem:

1. If we invest $100 or more on project 1, then we can only invest at most $100 on project 2:
This can be written as:
x ≥ 100 → y ≤ 100

If x is greater than or equal to 100, then y must be less than or equal to 100. This ensures that we don't invest more than $100 on project 2 if we invest $100 or more on project 1.

2. Investment amount cannot be negative:
This can be written as:
x ≥ 0
y ≥ 0

The investment amount on each project cannot be negative, so x and y must be greater than or equal to 0.

Therefore, the linear constraints for the problem are:
x ≥ 100 → y ≤ 100
x ≥ 0
y ≥ 0

First, let's define the variables:

Let x1 be the investment amount on project 1 (continuous variable)
Let x2 be the investment amount on project 2 (continuous variable)

Now, let's write the linear constraints based on the given condition:

If we invest $100 or more on project 1 (x1 ≥ 100), then we can only invest at most $100 on project 2 (x2 ≤ 100). To model this condition, we can use an integer variable:

Let y be an integer variable, with y ∈ {0, 1}

Now, we can write the linear constraints:

1. If y = 0, then x1 < 100 and there is no constraint on x2.
  y * 100 ≥ x1 (This ensures that if y = 0, x1 < 100)

2. If y = 1, then x1 ≥ 100 and x2 ≤ 100.
  (1 - y) * 100 ≥ x1 (This ensures that if y = 1, x1 ≥ 100)
  100y + x2 ≤ 200 (This ensures that if y = 1, x2 ≤ 100)

So the linear constraints are:
y * 100 ≥ x1
(1 - y) * 100 ≥ x1
100y + x2 ≤ 200

These constraints model the given condition, allowing you to analyze investments in both projects with continuous and integer variables.

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Answer:

10: Rectangular Prism

11: Pyramid

12: Triangular prism

Step-by-step explanation:

To understand subtraction as an unknown-addend problem means to view subtraction as finding the missing number in an addition problem.

In the example given, subtracting 10-8 means finding the number that needs to be added to 8 to make 10. This is essentially an addition problem with an unknown addend.

To solve the problem, one needs to think about what number added to 8 will result in 10. This can be done by counting up from 8 until you reach 10, which is a difference of 2. Therefore, 10-8=2, and the missing number is 2.

In general, to subtract any two numbers using this method, you can start with the smaller number and count up until you reach the larger number. The difference between the two numbers is the missing addend. For example, to subtract 15-7, you would start with 7 and count up to 15, which is a difference of 8. So 15-7=8.

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To find the equation of the line of intersection of two planes, we need to find the direction vector of the line. This can be done by taking the cross product of the normal vectors of the two planes.

Let the two planes be:
P1: 2x - y + 3z = 5
P2: x + 2y - 4z = -1
The normal vectors of these planes are:
n1 = <2, -1, 3>
n2 = <1, 2, -4>
Taking the cross product of these two vectors, we get:
n1 x n2 = <14, 10, 5>
This is the direction vector of the line of intersection.
To get the vector form of the equation, we need a point on the line. We can choose any point that lies on both planes. To make it easy, we can set z = 0 in both planes and solve for x and y.
From P1:
2x - y = 5
From P2:
x + 2y = -1
Solving these equations, we get:
x = -7/5
y = -3/5
So a point on the line is (-7/5, -3/5, 0).
Using this point and the direction vector, the vector form of the equation of the line of intersection is:
r = <-7/5, -3/5, 0> + t<14, 10, 5>
Here, t represents the parameter.

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Answer:

sqrt2(x+10)/2

Step-by-step explanation:

Answer: m∠CBD = 98°

Step-by-step explanation:

This shows that β^/r^ is an unbiased estimator of β/r.

True:

If β^ is a consistent estimator of β, it means that as the sample size increases, the estimator approaches the true value of the parameter β. Similarly, if r^ is a consistent estimator of r, then r^ approaches the true value of r as the sample size increases.

Using the algebraic property of limits, we can write:

lim β^/r^ = lim β^ / lim r^

As both β^ and r^ are consistent estimators, their limits exist and are equal to β and r respectively. Hence, we can write:

lim β^/r^ = β/r

This shows that β^/r^ is a consistent estimator of β/r.

True:

If β^ is an unbiased estimator of β, it means that the expected value of the estimator is equal to the true value of the parameter β. Similarly, if r^ is an unbiased estimator of r, then the expected value of r^ is equal to the true value of r.

Using the algebraic property of expected values, we can write:

E(β^/r^) = E(β^) / E(r^)

As both β^ and r^ are unbiased estimators, their expected values exist and are equal to β and r respectively. Hence, we can write:

E(β^/r^) = β/r

This shows that β^/r^ is an unbiased estimator of β/r.

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Lauren should invite at least 25 people to her party in order to have at least 23 guests.

Option D is the correct answer.

We have,

Let x be the number of people Lauren should invite to her party.

Since 92% of the people she invites are expected to come, the actual number of guests she can expect is 0.92x.

We want to find the value of x that makes the expected number of guests at least 23.

Therefore, we can set up the following inequality:

0.92x ≥ 23

Solving for x, we divide both sides by 0.92:

x ≥ 25

Therefore,

Lauren should invite at least 25 people to her party in order to have at least 23 guests.

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A. The 95% confidence interval is 0.291 < p < 0.435. B. The 80% confidence interval is (0.303, 0.397). C. A 99% confidence interval is 0.613 < p < 0.703.

A. Using the formula:

CI = p ± zsqrt(p(1-p)/n)

where p is the sample proportion, n is the sample size, and z is the critical value from the standard normal distribution. For a 95% confidence level, z is 1.96.

Putting the values:

CI = 131/396 ± 1.96sqrt((131/396)(265/396)/396)

Simplifying:

CI = 0.291 < p < 0.435

Therefore, the 95% confidence interval for the population proportion p is 0.291 to 0.435.

B. For an 80% confidence level, z is 1.282.

Putting the values:

CI = 0.35 ± 1.282sqrt((0.35)(0.65)/367)

Simplifying:

CI = (0.303, 0.397)

Therefore, the 80% confidence interval for the population proportion p is (0.303, 0.397).

C. For a 99% confidence level, z is 2.576.

Putting the values:

CI = 384/600 ± 2.576sqrt((384/600)(216/600)/600)

Simplifying:

CI = 0.613 < p < 0.703

Therefore, the 99% confidence interval for the population proportion p is 0.613 to 0.703. Writing it in tri-inequality form, we get:

0.613 < p < 0.703

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To find the critical value for a two-tail test with a sample standard deviation, an n-value of 36, and an alpha of 0.05, you will need to use the t-distribution table (as the population standard deviation is not given). However, since you have only provided the sample standard deviation and not its actual value, I cannot calculate the specific critical value for you.

Here are the general steps to find the critical value:
1. Calculate the degrees of freedom: df = n - 1 = 36 - 1 = 35
2. Divide the alpha (0.05) by 2 for a two-tail test, resulting in 0.025 in each tail.
3. Look up the t-value corresponding to the degrees of freedom (35) and the given alpha level (0.025) in the t-distribution table.
4. The t-value you find in the table is your critical value.

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The distance that the plane has flown, c, is 1,683 ft.

The distance of the plane is calculated by applying trigonometry ratio as shown below;

SOH CAH TOA

SOH = sin θ = opposite /hypothenuse side

TOA = tan θ = opposite side / adjacent side

CAH = cos θ = adjacent side / hypothenuse side

The height attained by the plane is the opposite side, while the hypothenuse is the distance travelled by the plane.

sin (12) = h/c

c = h/sin(12)

c = (350 ft ) / sin(12)

c = 1683.4 ft ≈1,683 ft

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The parameters of the sinusoidal function, y = -3·cos(π·(π - 2)) - 4, obtained from the equation of the function are;

(a) a) 2

b) 4 units down

c) 2 units left

(b) d) Please find attached the graph of the function showing the period created with MS Excel.

A sinusoidal function is a periodic sine or cosine based function.

The specified sinusoidal function can be presented as follows;

y = -3·cos(π·(x - 2)) - 4

The general form of a sinusoidal function is; y = A·cos(B·(x + C)) + D

(a) a) The period of a sinusoidal function is T = 2·π/|B|

A comparison with the general form of a sinusoidal function indicates;

A = 3, B = π, C = -2, D = -4

B = π

Therefore; T = 2·π/π = 2

The period, T = 2

b) The vertical shift of the function, D = -4

c) The horizontal shift of the function, C = -2

(b) d) Please find attached the graph of the function created with MS Excel

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Answer:

2/13

Step-by-step explanation:

The fee should the companyfee should the company charge to maximize their revenue charge to maximize their revenue is $7.50.

To determine the fee that the online computer game company should charge to maximize their revenue, we need to calculate the revenue for each fee rate listed in the table below:

| Fee Rate | Number of Subscribers | Revenue |
|----------|----------------------|---------|
| $8.00    | 10,000               | $80,000 |
| $7.75    | 10,500               | $81,375 |
| $7.50    | 11,000               | $82,500 |
| $7.25    | 11,500               | $83,375 |
| $7.00    | 12,000               | $84,000 |
| $6.75    | 12,500               | $84,375 |
| $6.50    | 13,000               | $84,500 |
| $6.25    | 13,500               | $84,375 |
| $6.00    | 14,000               | $84,000 |

From the table, we can see that the revenue initially increases as the fee rate decreases, but then starts to decrease after $7.50. This is because although the number of subscribers increases with a lower fee rate, the decrease in revenue from each individual subscriber outweighs the increase in subscribers.

Therefore, the fee rate that would maximize revenue for the online computer game company is $7.50. This is the fee rate where the revenue is the highest, at $82,500. We know this is the optimal fee rate because any higher fee rate will result in fewer subscribers, and any lower fee rate will result in a decrease in revenue per subscriber.

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The product of the fractions is 275/20

A fraction can simply be defined as the part of a given whole variable, a whole number or a whole element.

In mathematics, there are different types of fractions, namely;

From the information given, we have that;

To determine the product of the fraction

6 1/4×2 1/5

convert to improper fractions, we get;

25/4 × 11/5

Multiply the numerators

275/4(5)

Multiply the denominator

275/20

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As per the given volume, the new radius of the sphere is approximately 2.04 cm.

Let's first find the initial volume of the sphere. We know that the volume of a sphere is given by the formula V = (4/3)πr³, where V is the volume and r is the radius. Substituting the given values, we get:

V = (4/3)π(4³) = (4/3)π(64) = 268.08 cm³

Now, we can use the inverse proportionality between the volumes of the sphere and cone to find the new radius of the sphere. We know that the initial volume of the sphere (268.08 cm³) and the volume of the cone (48 cm³) are in inverse proportion to each other. This means that:

V(s) / V(c) = k

where k is a constant. We can find the value of k by substituting the initial volumes of the sphere and cone:

268.08 / 48 = k k = 5.584

Now, we can use this value of k to find the new radius of the sphere. We know that the new volume of the sphere is 6 cm³. This means that:

V(s) / V(c) = k

V(s) / 48 = 5.584

V(s) = 6 cm³

Substituting the values, we get:

6 / 48 = (4/3)πr³ / (4/3)π(4³)

Simplifying, we get:

r³ = 16/3

r = ∛(16/3)

r ≈ 2.04 cm

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we can choose k = (m + Hn-1)/(O(n^3 log n^3)) to obtain the desired upper bound on Pr[Y ≥ k].

Assuming there are i non-empty bins so far, the probability that the next ball you throw will fall into an empty bin is (n-i)/n. This is because there are n bins in total, and i of them are already non-empty. Therefore, there are (n-i) empty bins left, and each has an equal probability of receiving the ball.

To find E[Y], we can use linearity of expectation. Let Yi be the number of balls we need to throw to hit the next empty bin after i bins have been filled. Then, we have:

Y = Y1 + Y2 + ... + Yn

We know that Y1 = m, since we need to throw m balls to fill the first bin. For i > 1, we have:

E[Yi] = 1/(n-i+1) + E[Yi-1]

The first term represents the probability of hitting an empty bin on the next throw, and the second term represents the expected number of throws we need after that happens. We can solve this recurrence relation by substitution:

E[Y2] = 1/(n-1) + E[Y1] = 1/(n-1) + m

E[Y3] = 1/(n-2) + E[Y2] = 1/(n-2) + 1/(n-1) + m

E[Y4] = 1/(n-3) + E[Y3] = 1/(n-3) + 1/(n-2) + 1/(n-1) + m

...

We can see a pattern emerging, where each term has an additional 1/(n-i) compared to the previous term. Therefore, we have:

E[Y] = m + 1/(n-1) + 1/(n-2) + ... + 1/n

= m + Hn-1

≈ m + ln(n) (where Hn-1 is the (n-1)th harmonic number)

To show that E[Y] = O(n log n), we can use the fact that the harmonic series diverges, but the sum of the first n terms is bounded by ln(n) + 1. Therefore, we have:

E[Y] ≈ m + ln(n) ≤ m + ln(n) + 1

= m + O(log n)

Since m is a constant, we can say that E[Y] = O(n log n).

Using Markov's inequality, we have:

Pr[Y ≥ k] ≤ E[Y]/k

Therefore, we want to find k such that:

E[Y]/k ≤ O(n^3 log n^3)

Substituting in the expression we found for E[Y], we have:

(m + Hn-1)/k ≤ O(n^3 log n^3)

Rearranging, we get:

k ≥ (m + Hn-1)/(O(n^3 log n^3))

Therefore, we can choose k = (m + Hn-1)/(O(n^3 log n^3)) to obtain the desired upper bound on Pr[Y ≥ k].

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The requried, value is less than 8.5, as Imran claimed. Therefore, he is correct and not wrong.

To verify this, we can use the formula given:

Petrol consumption = 100 × litres of petrol used / kilometres travelled

Substituting the given values, we get:

Petrol consumption = 100 × 21.3 / 250 = 8.52

Rounding this value to three significant figures, we get:

Petrol consumption = 8.52

This value is indeed less than 8.5, as Imran claimed. Therefore, he is correct and not wrong.

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To find the probability of the event "the first die is 6 or the sum is 8," we need to count the number of outcomes that satisfy this event and divide it by the total number of possible outcomes.

To find the probability of the event "the first die is 6 or the sum is 8," we need to consider the total possible outcomes when rolling two dice and the favorable outcomes for this event.

Total possible outcomes: 6 sides on each die, so there are 6 x 6 = 36 possible outcomes.

Favorable outcomes:
1. First die is 6: There are 6 possible outcomes (6,1), (6,2), (6,3), (6,4), (6,5), and (6,6).
2. Sum is 8: There are 5 possible outcomes (2,6), (3,5), (4,4), (5,3), and (6,2).

However, (6,2) is counted twice, so we should subtract 1 from the total favorable outcomes: 6 + 5 - 1 = 10.

Probability = Favorable outcomes / Total possible outcomes = 10/36. Simplifying the fraction gives 5/18.

So, the probability of the event "the first die is 6 or the sum is 8" is 5/18.

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Answer:

(12, 28 )

Step-by-step explanation:

y = 2x + 4 → (1)

y = 3x - 8 → (2)

substitute y = 3x - 8 into (1)

3x - 8 = 2x + 4 ( subtract 2x from both sides )

x - 8 = 4 ( add 8 to both sides )

x = 12

substitute x = 12 into either of the 2 equations

substituting into (1)

y = 2(12) + 4 = 24 + 4 = 28

solution is (12, 28 )

The data provide convincing evidence that the pediatrician's claim is true

To test the pediatrician's claim that the mean weight of one-year-old boys is greater than 22 pounds, we can use a one-sample t-test.

Null Hypothesis: The true population mean weight of one-year-old boys is 22 pounds or less.

Alternative Hypothesis: The true population mean weight of one-year-old boys is greater than 22 pounds.

We can use a level of significance of 0.05, which corresponds to a 95% confidence level.

The test statistic for this one-sample t-test is:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

Substituting the given values:

t = (23.5 - 22) / (2.8 / sqrt(315)) = 10.15

Using a t-distribution table with 314 degrees of freedom (n-1), we find that the critical value for a one-tailed test at the 0.05 level of significance is 1.646.

Since the calculated t-value (10.15) is greater than the critical value (1.646), we reject the null hypothesis.

Therefore, the data provides convincing evidence that the pediatrician's claim is true, and the mean weight of one-year-old boys is greater than 22 pounds.

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We can state the critical values for the rejection rule as follows:

test statistic ≤ -1.645 (left-tailed test)

test statistic ≥ 1.645 (right-tailed test)

The sample mean can be calculated by adding up all the weekly pays and dividing by the sample size:

sample mean = (687.73 + 543.15 + ... + 703.06) / 50 = 638.55 (rounded to two decimal places)

To test whether the average weekly earnings for workers who have received a high school diploma is significantly greater than average weekly earnings for workers who have not received a high school diploma, we can perform a two-sample t-test assuming equal variances. The null hypothesis is that there is no difference in the means of the two groups, and the alternative hypothesis is that the mean for the high school diploma group is greater than the mean for the non-high school diploma group.

Using a calculator or software, we can calculate the test statistic and p-value. Assuming a two-tailed test and a significance level of 0.05, the critical values for the rejection rule are -1.96 and 1.96.

test statistic = 3.196 (rounded to three decimal places)

p-value = 0.0012 (rounded to four decimal places)

Since the p-value (0.0012) is less than the significance level (0.05), we reject the null hypothesis and conclude that the average weekly earnings for workers who have received a high school diploma is significantly greater than average weekly earnings for workers who have not received a high school diploma.

For a one-tailed test with α = 0.05, the critical value would be 1.645. The rejection rule would be: if the test statistic is greater than 1.645, reject the null hypothesis. Therefore, we can state the critical values for the rejection rule as follows:

test statistic ≤ -1.645 (left-tailed test)

test statistic ≥ 1.645 (right-tailed test)

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The accumulated balance after 17 years is $18,481.76.

What is compound interest?

Compound interest is when you earn interest on both the money you've saved and the interest you earn.

To solve this problem, we can use the formula for the future value of an annuity:

[tex]FV = PMT * [(1 + r/n)^{(n*t) - 1]} / (r/n)[/tex]

where FV is the future value, PMT is the periodic payment, r is the interest rate, n is the number of compounding periods per year, and t is the number of years.

In this case, PMT = $146.30, r = 5.3%, n = 4 (since interest is compounded quarterly), and t = 17. Plugging these values into the formula, we get:

[tex]FV = $146.30 * [(1 + 0.053/4)^{(4*17)} - 1] / (0.053/4)[/tex]

[tex]= $146.30 * (1.01325^{68} - 1) / 0.01325[/tex]

= $146.30 * 126.3584

= $18,481.76

Therefore, the accumulated balance after 17 years is $18,481.76.

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The value ox in each of the equations given are:

1. x = 3        2. x = 5

For each of the equation given, we can solve to find the value of x by isolating the variable to one side of the equation while applying the properties of equality.

1. 2x + 5 = 11

2x = 11 - 5 [subtraction property of equality]

2x = 6

x = 6/2 [division property]

x = 3

2. 3x - 6 = 9

3x = 9 + 6 [addition property]

3x = 15

3x/3 = 15/3 [division property]

x = 5

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The length of the rope is approximately 13.1 feet. (option a).

The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. In this case, the adjacent side is the part of the rope that is attached to the pole, and the hypotenuse is the length of the rope.

Using the cosine function, we have:

cos(40) = adjacent side / hypotenuse

Rearranging this equation, we get:

hypotenuse = adjacent side / cos(40)

The adjacent side is the length of the part of the rope that is attached to the pole, which is 10 feet. Therefore, we can substitute this value and the angle into the equation to get:

hypotenuse = 10 / cos(40)

Using a calculator, we can find that cos(40) is approximately 0.766. Therefore, we have:

hypotenuse = 10 / 0.766

Simplifying this expression, we get:

hypotenuse ≈ 13.1 feet

Hence the correct option is (a).

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If the product of the terms of the fraction is 162 and the fraction is equivalent to 2/9 then the fraction is 6/27.

Let's assume the two terms of the fraction are x and y. We are given that xy = 162. Also, we know that the fraction is equivalent to 2/9. So, we can write,

x/y = 2/9

Cross-multiplying, we get,

9x = 2y

Solving for y, we get,

y = (9/2)x

Substituting this value of y in the equation xy = 162, we get,

x(9/2)x = 162

Simplifying, we get,

(9/2)x² = 162

x² = 36

x = 6 (taking the positive square root as x and y are positive)

Substituting this value of x in the equation y = (9/2)x, we get,

y = (9/2)(6) = 27

So, the fraction is 6/27, which simplifies to 2/9.

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Complete question - The product of two terms of a fraction is 162, find the fraction if it is equivalent to 2/9.

This indicates that the particle is accelerating due to gravity. At t=7, the rate at which the velocity is changing depends on the value of k, m, and g. This implies that the particle's acceleration may vary depending on these constants and time.

At time 0, the rate at which the velocity is changing can be found by taking the derivative of the velocity function with respect to time, t.

v(t) = (mg/k)(1-e^(-kt/m))

v'(t) = (mg/k)((ke^(-kt/m))/m)

Plugging in t=0 gives:

v'(0) = (mg/k)((k)/m) = g

Therefore, the rate at which the velocity is changing at time 0 is g.

At t=7, the rate at which the velocity is changing can also be found by taking the derivative of the velocity function with respect to time, t, and plugging in t=7:

v'(7) = (mg/k)((ke^(-7k/m))/m)

This value will depend on the specific values of k, g, and m that are not given in the question.

These rates of change tell us about the motion of the particle. If the rate of change of velocity is positive, the particle is accelerating. If it is negative, the particle is decelerating. If it is zero, the particle is moving at a constant velocity.

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The set {1, x, x²} is a basis for the subspace of monic polynomials of degree at most 2 with real coefficients.

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

i) The subspace spanned by a vector v = [a,b] in R2 is the set of all scalar multiples of v. In other words, it is the line passing through the origin and the point (a,b).

So, the subspace spanned by v can be written as:

Span(v) = {k[a,b] : k ∈ R}

where k is a scalar. Note that [a,b] is the basis for this subspace.

ii) The set of monic polynomials of degree at most 2 with real coefficients is:

{1, x, x²}

This set spans a subspace of the vector space of all real polynomials of degree at most 2. To see this, let p(x) be an arbitrary polynomial of degree at most 2 with real coefficients. Then we can write:

p(x) = ax² + bx + c

where a, b, and c are real numbers. Now, we can express p(x) as a linear combination of the monic polynomials:

p(x) = a(x²) + b(x) + c(1)

Therefore, any polynomial of degree at most 2 with real coefficients can be written as a linear combination of the monic polynomials.

To find a basis for this subspace, we need to determine which of these monic polynomials are linearly independent. One way to do this is to see if any of the monic polynomials can be expressed as a linear combination of the others. In this case, it is clear that none of the monic polynomials can be expressed as a linear combination of the others.

Therefore, the set {1, x, x²} is a basis for the subspace of monic polynomials of degree at most 2 with real coefficients.

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