Find a polynomial function whose graph passes through each set of points.
a. (-3, 15), (1, 11), and (0, 6)

b. (-2,-7), (-1, -3), (0, 3), (1, 5), and (2, -3)

c. (4,-1) and (-3, 13)

d. (-1,-6), (0, 2), (1, 8), and (2, 42)

Thank you!!

The complete table:

Number of paces per minute      Pace length (metres)

63                                                            0.45

70                                                             0.5

77                                                             0.55

84                                                            0.6

To fill in the missing numbers in the table, we can use the relationship given as:

n / P = 140

Where n is the number of paces per minute, and P is the pace length in meters.

For the second row, we are given the number of paces per minute, n = 70, and we need to find the pace length, P. We can rearrange the formula as:

P = n / 140

Substituting the values, we get:

P = 70 / 140 = 0.5 meters

So the missing pace length in the table is 0.5 meters.

For the fourth row, we are given the pace length, P = 0.6 meters, and we need to find the number of paces per minute, n. We can rearrange the formula as

n = P x 140

Substituting the values, we get:

n = 0.6 x 140 = 84

So the missing number of paces per minute in the table is 84.

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The limit along the line y = x is different from the limits along the x-axis and y-axis, the limit as (x, y) approaches (0, 0) does not exist for the given expression.

To find the limit as (x, y) approaches (0, 0) of the expression y^2 sin(x^2) / (x^4 y^4), we can analyze the limit along different paths and see if they converge to the same value. If they do not, the limit does not exist.

Let's consider the limit along the x-axis first, where y = 0:

lim(x->0) [0^2 sin(x^2) / (x^4 * 0^4)] = 0.

Next, let's consider the limit along the y-axis, where x = 0:

lim(y->0) [y^2 sin(0^2) / (0^4 * y^4)] = 0.

Now, let's examine the limit along the line y = x:

lim(x->0) [x^2 sin(x^2) / (x^4 * x^4)] = lim(x->0) [sin(x^2) / x^6].

By applying L'Hôpital's rule repeatedly, we can find the limit:

lim(x->0) [sin(x^2) / x^6] = lim(x->0) [2x cos(x^2) / 6x^5] = lim(x->0) [2 cos(x^2) / 6x^4] = lim(x->0) [cos(x^2) / 3x^4] = lim(x->0) [(-2x sin(x^2)) / (12x^3)] = lim(x->0) [(-2 sin(x^2)) / (12x^2)] = lim(x->0) [(-4x cos(x^2)) / (24x)] = lim(x->0) [(-4 cos(x^2)) / 24] = (-4/24) = -1/6.

Since the limit along the line y = x is different from the limits along the x-axis and y-axis, the limit as (x, y) approaches (0, 0) does not exist for the given expression.

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Mike saved his money for 7 years to earn $280 in interest at a simple interest rate of 2%.

The simple interest formula:

I = P × r × t

Where:

I is the interest earned

P is the principal (the initial amount of money saved)

r is the interest rate

t is the time (in years)

We know that Mike saves $2000 at a simple interest rate of 2% and he earns $280 in interest.

So we can plug in these values and solve for "t":

280 = 2000 × 0.02 × t

Dividing both sides by (2000 × 0.02):

280 / (2000 × 0.02) = t

t = 7

I = P r t is the formula for calculating interest.

P stands for principle, which is the original sum of money saved and r stands for interest rate.

The date is (in years).

We are aware that Mike gets $280 in interest on his savings of $2000 at a basic interest rate of 2%.

Thus, we may enter these numbers and find the value of "t":

280 = 2000 × 0.02 × t

by (2000 0.02), divide both sides:

280 / (2000 × 0.02)= 7

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The multimedia element that would enhance a speech is multimedia element would enhance a speech about the problems of coastal erosion, the correct option is B.

We are given that;

The four options

Now,

A multimedia element is a visual or auditory aid that can enhance a speech by making it more engaging, informative or persuasive. A multimedia element should be relevant to the topic, clear and accurate, and appropriate for the audience and occasion. Here are some criteria to evaluate the multimedia elements:

A road that has collapsed next to a beach. This element is relevant to the topic of coastal erosion, as it shows one of the consequences of losing land and infrastructure due to erosion. It is also clear and accurate, as it depicts a realistic scenario that might happen in some areas. It might engage and persuade the audience by appealing to their emotions or interests.

Therefore, by the unitary method answer will be A road that has collapsed next to a beach.

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The estimated mouse population after 5 years is 7,680 mice. (a) The mouse population is doubling every year,

which means that the population at any time t will be double the population at time t-1. We can use this information to write the function N(t) that models the number of mice after t years as follows: N(t) = 240 * 2^t

(b) To estimate the mouse population after 5 years, we can simply substitute t=5 into the function we found in part (a): N(5) = 240 * 2^5 = 7,680

Therefore, the estimated mouse population after 5 years is 7,680 mice. However, it is important to note that this is only an estimate based on the assumption that the population is doubling every year.

In reality, there may be factors such as limited resources and predation that could affect the growth rate of the population.

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The probability of drawing a card that is both odd and divisible by 4 from a deck of 16 cards numbered 1 through 16 is 0.

To see why, note that a number that is both odd and divisible by 4 does not exist in the given set of numbers. Every even number in the set is divisible by 2 but not 4, while every multiple of 4 in the set is even and not odd. Therefore, the intersection of the sets of odd and multiples of 4 is the empty set, and the probability of drawing a card that belongs to this set is 0.

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The curve has horizontal tangents at the points (0,0) for all values of t. To find the points at which the curve has a horizontal tangent, we need to find the values of t that make the derivative of y(t) equal to zero.

First, we need to find the derivative of y(t):

y'(t) = 8cos(t)

Next, we set y'(t) equal to zero and solve for t:

8cos(t) = 0

cos(t) = 0

This occurs when t = π/2 or 3π/2.

Now, we can plug these values of t back into the original equations to find the corresponding points:

When t = π/2,

x(π/2) = 4cos(π/2) = 0

y(π/2) = 4sin(2(π/2)) = 0

So the point is (0,0).

When t = 3π/2,

x(3π/2) = 4cos(3π/2) = 0

y(3π/2) = 4sin(2(3π/2)) = 0

So the point is also (0,0).

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Step-by-step explanation:

To find f(t), we need to take the inverse Laplace transform of 1/(s^2 - 4s + 5).

We can start by factoring the denominator of the Laplace transform:

1/(s^2 - 4s + 5) = 1/[(s - 2)^2 + 1^2]

We can recognize this as the Laplace transform of the function f(t) = e^2t * sin(t). Therefore,

ℒ^{-1} {1/(s^2 - 4s + 5)} = e^{2t} sin(t)

Thus, f(t) = e^{2t} sin(t).

There are no relative maximum or minimum points for the function f (x, y) = x^2y^2 - 18x - 8y - 5.

To find the relative maximum and minimum of the function f (x, y) = x^2y^2 - 18x - 8y - 5, we need to find the critical points and classify them using the second derivative test.

First, we find the partial derivatives of f with respect to x and y:

f x (x, y) = 2xy^2 - 18

f y (x, y) = 2x^2y - 8

To find the critical points, we set the partial derivatives to zero and solve for x and y:

2xy^2 - 18 = 0

2x^2y - 8 = 0

Solving these equations simultaneously, we get:

x = ±√9y^2

y = ±√2

So the critical points are:

(3√2, √2)

(3√2, -√2)

(-3√2, √2)

(-3√2, -√2)

To classify these critical points, we need to find the second partial derivatives:

f x x (x, y) = 2y^2

f y y (x, y) = 2x^2

f x y (x, y) = 4xy

Then, we evaluate the second partial derivatives at each critical point:

f x x (3√2, √2) = 4

f y y (3√2, √2) = 18

f x y (3√2, √2) = 12√2

f x x (3√2, -√2) = 4

f y y (3√2, -√2) = 18

f x y (3√2, -√2) = -12√2

f x x (-3√2, √2) = 4

f y y (-3√2, √2) = 18

f x y (-3√2, √2) = -12√2

f x x (-3√2, -√2) = 4

f y y (-3√2, -√2) = 18

f x y (-3√2, -√2) = 12√2

At each critical point, we have:

D = f x x (x, y) * f y y (x, y) - f x y (x, y)^2 = (4) (18) - (12√2)^2 = -288 < 0

Since the discriminant D is negative at each critical point, we can conclude that f has a saddle point at each critical point.

Therefore, there are no relative maximum or minimum points for the function f (x, y) = x^2y^2 - 18x - 8y - 5.

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The bearing form D from A, according to the figure is

205 degrees

Bearings are measured form the North and in the clockwise direction

Examining the figure and applying the clockwise direction to measure the angles, we have the bearing as

bearing form D from A = angle N to B + angle B to C + angle C to D

angle C to D is not given and solved using sum of angles in a point

angle C to D = 360 - 155 - 15 - 166

angle C to D = 24 degrees

plugging in the values

bearing form D from A = 15 + 166 + 24

bearing form D from A = 205 degrees

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It will take 7 weeks for the number of bacteria cells to double enough times and reach 16,000 cells

Now, By using an exponential growth model to find the time it would take for the number of bacteria cells to reach 16,000:

N(t) = N₀ * 2^(t/k)

where N₀ is the initial number of bacteria cells, N(t) is the number of bacteria cells at time t, k is the doubling time (in weeks), and t is the time (in weeks).

We know that;

N₀ = 125,

And, For the value of t when N(t) = 16,000.

So we can set up an equation:

16,000 = 125 x 2^(t/k)

Next, we can simplify this equation by dividing both sides by 125:

128 = 2^(t/k)

Then we can take the logarithm of both sides (with base 2) to solve for t/k:

log2(128) = t/k

7 = t/k

Therefore, it will take 7 weeks for the number of bacteria cells to double enough times and reach 16,000 cells.

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[tex]E=\{0,1,2,3,4,5\}[/tex]

The remainders from dividing a natural number by a natural number [tex]n[/tex] are [tex]0,1,2,3,\ldots,n-1[/tex].

Any values of m that satisfy this equation will also satisfy the inequality m - 3 > -2. However, this equation has infinitely many solutions, since we can choose any value for k and get a corresponding value for m.

The inequality m - 3 > -2 can be solved as follows:

Add 3 to both sides of the inequality to isolate the variable m:

m - 3 + 3 > -2 + 3

m > 1

So the equivalent inequality is: m > 1.

Alternatively, we could also write the inequality in the form of an equation by adding a variable k on both sides of the inequality:

m - 3 + k = -2 + k

Simplifying this equation, we get:

m = k - 1

So any values of m that satisfy this equation will also satisfy the inequality m - 3 > -2. However, this equation has infinitely many solutions, since we can choose any value for k and get a corresponding value for m.

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Answer: The best prediction possible for the number of times you roll a three is 1/6 of a roll.

Step-by-step explanation:

When you find the statistics of something like rolling a die, you will need to divide how many times you roll the die by how many sides are on the die.

6 - how many times you roll the die

6 - how many sides are on the die

When you divide 6 by 6,  you will get one. Since both of the statistical numbers are the same, you will get an 1/6 chance of rolling a 1, 2, 3, 4, 5, and 6. So 1/6 is the answer! Hope this helps!!!

P.S (I learned this in 5th grade :))

There will be 10 confidence intervals obtained in a Tukey multiple comparison of 5 populations.

In a Tukey multiple comparison, the confidence intervals are constructed to compare the means of all pairs of groups. To calculate the number of confidence intervals, we use the following formula:

C = n(n-1)/2

Where C is the number of confidence intervals, and n is the number of groups. In this case, there are five populations being compared, so n=5. Plugging this into the formula, we get:

C = 5(5-1)/2 = 10

Each confidence interval will provide information about the difference between the means of two groups, with a certain level of confidence. These confidence intervals can be used to identify which pairs of groups have significantly different means.

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The new surface area of the figure is given as follows:

S = 164.61 ft².

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The prism in this problem is composed as follows:

Hence the surface area of the original prism is given as follows:

S = 2 x (6.5 x 5.5 + 5.5 x 10.6 + 6.5 x 10.6) + 2 x (4.3 x 10.6 + 4.3 x 8.1 + 10.6 x 8.1)

S = 658.44 ft².

The new surface area is one fourth of the original surface area, hence it is given as follows:

S = 0.25 x 658.44

S = 164.61 ft².

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The coordinate matrix of x in rn relative to the basis b' is [-3, 2] because x can be expressed as -3 times the first vector in b' plus 2 times the second vector in b'.

To find the coordinate matrix of x in rn relative to the basis b', we need to express x as a linear combination of the basis vectors in b', and then write the coefficients of that linear combination as entries in the coordinate matrix. In this case, we have:

x = (-31, 37)

b' = {(-7, 8), (4, -3)}

We want to find scalars a and b such that:

x = a(-7, 8) + b(4, -3)

Solving this system of equations, we get:

a = -3

b = 2

Therefore, the coordinate matrix of x in rn relative to the basis b' is:

[-3, 2]

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Answer:

P(less than 2) = P(1) = 1/6

A is correct.

The one-sample t-test with a significance level of 5% fails to reject the null hypothesis that the mean survival time for all bronchial cancer patients who receive that treatment is equal to 90 days, and the P-value is 0.0974.

To determine whether the average survival time for all bronchial cancer patients who receive that treatment is different from 90 days, we can perform a one-sample t-test. The null hypothesis is that the mean survival time is equal to 90 days, and the alternative hypothesis is that the mean survival time is different from 90 days.

We can use the following formula to calculate the t-statistic:

[tex]$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$[/tex]

Substituting the given values, we get:

t = (111.3 - 90) / (62.9 / √12) = 1.83

Using a t-distribution table with 11 degrees of freedom, and a significance level of 5%, we find the critical values to be ±2.201.

Since the calculated t-value of 1.83 falls within the range of -2.201 to +2.201, we fail to reject the null hypothesis. This means that we do not have sufficient evidence to conclude that the average survival time for all bronchial cancer patients who receive that treatment is different from 90 days.

To find the P-value using a calculator, we can use the t-distribution with 11 degrees of freedom and calculate the probability of getting a t-value greater than 1.83 or less than -1.83 (since it is a two-tailed test). The P-value turns out to be 0.0974. Rounded to the nearest 4th decimal digit, the P-value is 0.0974.

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Complete question:

Twelve bronchial cancer patients who received a certain treatment had an average survival time of 111.3 days with a standard deviation of 62.9 days. Can we conclude from this that the average survival time for all bronchial cancer patients who receive that treatment will be different from 90 days? We will use a 5% significance level. Do the T-test on your calculator and enter the P-value below. Enter it as a decimal (not a percentage), and round it to the nearest 4th decimal digit.

To determine if distance can be an instrumental variable (IV) for attendance in the model of given performance, we need to ensure that it satisfies the following assumptions: Relevance, Exogeneity and Exclusion restriction.

1. Relevance: Distance must be correlated with attendance, meaning that it has a significant effect on attendance. Intuitively, students living closer to the lecture hall may attend classes more frequently.
2. Exogeneity: Distance must not be directly correlated with the error term (u) in the performance equation, meaning that it should not have any direct effect on student performance apart from its impact on attendance. The assumption given already states that distance and u are uncorrelated, which fulfills this requirement.
3. Exclusion restriction: Distance should not have any direct effect on performance except through its influence on attendance. In other words, after controlling for attendance, ACT scores, and GPA, distance should not be a significant predictor of performance.
Additionally, we must assume that there are no other unobserved variables that could be driving the relationship between distance and performance. If these assumptions are met, we can use distance as an instrumental variable to identify the causal effect of attendance on performance.

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Answer:

y = 0.0633 x^2

Step-by-step explanation:

Since the vertex of the parabola is at the origin, the equation of the parabola can be written in the form:

y = a x^2

where a is a constant that determines the shape of the parabola.

The focus of the parabola is at the point (0,-7.9). Recall that the focus of a parabola is a point that is equidistant from the vertex and the directrix. Since the vertex is at the origin, the directrix must be a horizontal line that is 7.9 units above the vertex. Therefore, the equation of the directrix is:

y = 7.9

The distance between the vertex and the focus is equal to the distance between the vertex and the directrix. This distance is given by:

d = |-7.9 - 0|/2 = 3.95

Therefore, the constant a can be found by solving the equation:

a = 1/(4d) = 1/(4(3.95)) = 0.0633

So the equation of the parabola is:

y = 0.0633 x^2

The sequence given by an = n^2 - 3n + 1, with n starting from 1, produces the values 0, 0, 2, 6, 12, 20, .... This sequence has a closed formula of a_n = (n-1)n, where n is a positive integer.

To find the first few terms of the sequence, we substitute the values of n starting from 1 into the formula an = n^2 - 3n + 1. This gives us the sequence: 0, 0, 2, 6, 12, 20, ....

To find the closed formula for the sequence, we can observe that each term is the sum of the previous term and n(n-1). In other words, a_n = a_(n-1) + n(n-1), where a_1 = 0. We can then use this recursive definition to find a closed formula. We start by finding the first few terms: a_1 = 0, a_2 = 2, a_3 = 6, a_4 = 12, a_5 = 20, ...

We notice that the difference between consecutive terms is n(n-1), so we can express the nth term in terms of the first term and the sum of n(n-1) from k=1 to n-1:

a_n = a_1 + ∑(k=1 to n-1) k(k-1)

Simplifying the sum using the formula for the sum of the first n integers and the sum of the first n squares, we get:

a_n = (n-1)n

Therefore, the closed formula for the sequence starting with a1 is a_n = (n-1)n, where n is a positive integer.

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b. Central Limit Theorem. in this scenario, we would use the Central Limit Theorem to determine the probability that the mean life expectancy will be less than 70 years.

The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution. In this case, a sample of 50 people is selected, which is reasonably large.

Since the mean and standard deviation of the population are known, we can calculate the z-score for a sample mean of 70 years. The z-score is calculated as:

z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

= (70 - 72) / (5.3 / sqrt(50))

≈ -1.19

By referring to a standard normal distribution table or using a calculator, we can find the probability of a z-score less than -1.19. This probability corresponds to the probability that the sample mean will be less than 70 years.

Using the Central Limit Theorem, we can approximate the distribution of the sample mean as a normal distribution and calculate the desired probability.

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If there is no variability (all the scores of the variable have the same value), measures of dispersion will equal 0. Option A) 0 is the correct answer.

When there is no variability, it means all the scores of the variable have the same value. Therefore, there is no difference between the scores, and the deviation of each score from the mean is zero. As a result, all measures of dispersion such as the range, variance, and standard deviation will be zero because they are all based on the deviation of scores from the mean.

Therefore, option A) 0 is the correct answer.

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The researchers estimate for the proportion of patients with high BMD in the population is equal to the sample proportion since they used a random sampling method with replacement. Therefore, their estimate for the proportion of patients with high BMD in the population is 0.267 or 26.7%.

In statistics, a population refers to the entire group of individuals, objects, or events that meet certain criteria. A sample, on the other hand, is a smaller subset of the population that is selected to represent the larger group.
In this case, the population is comprised of 622 patients, of whom 147 have high bone mineral density (BMD). The researchers selected a sample of 30 subjects from this population with replacement. This means that after selecting a subject, they put it back in the population before selecting the next subject.
Of the 30 subjects in the sample, 8 have high BMD. To estimate the proportion of patients with high BMD in the population, the researchers use the sample proportion.
The sample proportion is calculated by dividing the number of subjects with high BMD in the sample (8) by the total number of subjects in the sample (30).
Sample proportion = 8/30 = 0.267
It's important to note that the larger the sample size, the more accurate the estimate will be. However, in this case, the sample size is relatively small, so there may be some error in the estimate. Nonetheless, the researchers estimate provides a good indication of the proportion of patients with high BMD in the population.

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Answer:

radius is 6 meters

Step-by-step explanation:

perimeter of semicircle = πr + 2r

πr + 2r = 30.84

3.14r + 2r = 30.84

5.14r = 30.84

r = 30.84 ÷ 5.14

r = 6

The figures that Jamal could be constructing are: Circumcenter and circumscribed circle

- Centroid is the point of intersection of all the three medians.

- Circumcenter is the point where the perpendicular bisectors of the sides of that particular triangle intersect.

- Incenter of a triangle is the intersection point of all the three interior angle bisectors of the triangle

- Orthocenter is the point where all three altitudes of the triangle intersect.

- Center of mass is also the centroid of a triangle which is the intersection of the three medians of the triangle.

- For the circumscribed circle, we will need the perpendicular bisectors

Thus, only Circumcenter and circumscribed circle are correct.

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Answer:

b=35

Step-by-step explanation:

We are given that:

b=4a-5

and are asked to find b when a=10

So, we can first substitute in 10 for a:

b=4(10)-5

simplify

b=40-5

b=35

So, b=35.

Hope this helps! :)

Edit: There is not a c variable?  I'm confused so I left this out.

The phrase :  the difference between 17m and 3 can be written as 17m - 3.

We have,

Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

Given is a phrase : the difference between 17m and 3.

We can write the given phrase as -

17m - 3

Therefore, the phrase :  the difference between 17m and 3 can be written as 17m - 3.

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complete question:

What is the algebraic expression for the following word phrase: the difference between 17m and 3?

A. 17m + 3

B. 17m • 3

C. 17m ÷ 3

D. 17m - 3

Hi help me please