Can someone tell me what that n symbol next to pi is. What is it called? Picture attached.

def longest_palindromic_substring(s):
   def is_palindrome(sub):
       return sub == sub[::-1]
   
   n = len(s)
   if n == 0:
       return ""
   
   ans = s[0]
   

To solve this problem, we can use the following approach:

1. Create a function to check whether a given substring is a palindrome or not. We can do this by comparing the first and last characters of the substring and then moving towards the center until we have checked all the characters.

2. Start with a substring of length one and check if it is a palindrome. If it is, we update our answer to be this substring.

3. Then we expand our substring by adding one character to the left and one to the right. We check if this new substring is a palindrome, and if it is, we update our answer to be this substring.

4. We keep expanding our substring until we reach the end of the string. At each step, we check if the substring is a palindrome and update our answer accordingly.

5. Finally, we return the longest palindrome substring we found.

Here's the Python code for this approach:

```
def longest_palindromic_substring(s):
   def is_palindrome(sub):
       return sub == sub[::-1]
   
   n = len(s)
   if n == 0:
       return ""
   
   ans = s[0]
   
   for i in range(n):
       for j in range(i+1, n+1):
           sub = s[i:j]
           if is_palindrome(sub) and len(sub) > len(ans):
               ans = sub
               
   return ans
```

This solution has a time complexity of O(n^3) because we are checking all possible substrings of the input string. However, we can optimize this solution by using dynamic programming or Manacher's algorithm to reduce the time complexity to O(n^2) or O(n), respectively.

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The sweater is 50% off.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100.

To calculate the percentage off the sweater is, we can use the following formula:

Percentage Off = (Original Price - Sale Price) / Original Price * 100

In this case, the original price of the sweater is $70 and the sale price is $35.

Percentage Off = (70 - 35) / 70 * 100 = 50%

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The theorem described of the point of tangency would be the "Perpendicular Tangent Theorem.

If a tangent and diameter intersect at the point of tangency, then according to the Perpendicular Tangent-Diameter Theorem, they create a perpendicular. To prove this theorem, it is important to consider that every tangent line touches the circle's radius at a 90-degree angle at the point of intersection.

Thus, when a diameter passes through the center of the circle, it becomes a special type of radius - one which intersects with two points on the circumference instead of just one. At each point where it intercepts the circumference, we see a right-angle between the tangent line and the diameter, as required by the Perpendicular Tangent-Diameter Theorem.

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Answer:

(1, 1/2)

Step-by-step explanation:

To find the midpoint of a line segment with endpoints (x1,y1) and (x2,y2), we can use the midpoint formula which is ((x1+x2)/2, (y1+y2)/2) .

In this case, A has coordinates (3,-2) and B has coordinates (-1,3). Plugging these values into the midpoint formula gives us:

((3 + (-1))/2, (-2 + 3)/2) = (1, 1/2)

Therefore, the midpoint of AB has coordinates (1, 1/2) . I hope that helps!

The importance of the standard normal distribution arises from the fact that any normal random variable can be transformed or standardized into the standard normal random variable to derive the relevant probabilities.

The normal distribution is a continuous probability distribution used to model random variation around the mean of a variable.The importance of the standard normal distribution arises from the fact that any normal random variable can be transformed. It is characterized by two parameters: the mean (μ) and the standard deviation (σ). The standard normal distribution is a special case with a mean of zero and a standard deviation of one, denoted by Z. It has a bell-shaped curve, and the area under the curve between two values represents the probability of observing a value in that interval.

To transform a normal distribution into the standard normal distribution, standardization is employed. This process involves subtracting the mean of the normal distribution from the random variable and dividing the result by the standard deviation. The resulting standardized value has a mean of zero and a standard deviation of one, following the formula: z = (x - μ) / σ.

Once the standard normal distribution is obtained, probabilities can be calculated using the z-table. This table provides the area under the curve of the standard normal distribution to the left of a given value of z. By looking up the relevant value in the z-table, the probability of observing a value less than or equal to that z-value can be determined.

For example, let's consider a normal distribution with a mean of 70 and a standard deviation of 10. To find the probability of observing a value less than 80, we standardize the value as follows: z = (80 - 70) / 10 = 1. We then consult the z-table and find the area under the curve to the left of z = 1, which is 0.8413. Therefore, the probability of observing a value less than or equal to 80 in the original normal distribution is 0.8413.

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The probability that z lies between -1.12 and 2.73 in a standard normal distribution is approximately 0.8653.

Let z be a random variable with a standard normal distribution. You need to find the probability P(-1.12 ≤ z ≤ 2.73).
To find this probability, follow these steps:
1. Identify the given values: z1 = -1.12 and z2 = 2.73.
2. For a standard normal distribution, you need to use the standard normal table (z-table) or a calculator with a built-in standard normal distribution function to find the probabilities associated with z1 and z2.
3. Look up the probabilities for z1 and z2 in the z-table or use a calculator:
  P(z ≤ -1.12) = 0.1314 and P(z ≤ 2.73) = 0.9967.

4. Calculate the probability of the given range by subtracting the smaller probability from the larger one:
  P(-1.12 ≤ z ≤ 2.73) = P(z ≤ 2.73) - P(z ≤ -1.12) = 0.9967 - 0.1314 = 0.8653.
5. Round your answer to four decimal places:
  P(-1.12 ≤ z ≤ 2.73) = 0.8653.
So, the probability that z lies between -1.12 and 2.73  is approximately 0.8653.

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This exponential function captures the exponential decay of the number of players as the tournament progresses.

The exponential function that can be used to find the number of players left in the tournament during each round is:

N(t) = N(0) * (1/2)^t

Where:

N(t) represents the number of players remaining after t rounds of tennis.

N(0) is the initial number of participants at the start of the tournament, which is 128 in this case.

t represents the number of rounds played.

The function (1/2)^t represents the fact that during each round, half of the players are eliminated. Each time a round is played, the remaining number of players is halved.

For example, after 1 round, the number of players remaining is N(1) = 128 * (1/2)^1 = 64. After 2 rounds, N(2) = 128 * (1/2)^2 = 32, and so on.

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Answer:

A(x) = (x/4)^2 + ((10 - x)/4)^2

A'(x) = 2(x/4)(1/4) + 2((10 - x)/4)(-1/4)

x/8 + (x - 10)/8 = 0

x + x - 10 = 0

2x = 10, so x = 5

Cut the 10-meter wire into two 5-meter pieces. The area of each square is (5/4)^2 = 25/16 = 1.5625 square meters, so the combined area for both squares is 25/8, or 3.125 square meters (1.5625 square meters per square).

The domain of the function is the set of all real values and the range of the function is y >= 5

From the question, we have the following parameters that can be used in our computation:

The graph of the quadratic function

As a general rule

The domain of a quadratic function is the set of all real values

For the range, we have

The graph has a minimum value at y = 5

Also, the values of the function output (y values) increases

This means that the range is y >= 5

Hence the domain is the set of all real values and the range is y >= 5

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The equation that represents the data given (line r and s) is  y = (-19/5)x + 76/5.; y = 5y-8 respectively.

To get the equations of the lines represented by the given tables, we can use the point-slope form of the equation of a line...

y - y₁  = m ( x - x₁)

where

( x₁, y ₁) is a point on the line, and

m is the slope of the line.

For line r, we can use points (4 ,  8) and (9, -11) to find the slope ...

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

=

(-11 - 8) / (9 - 4) = - 19/5

Let's use the point (4, 8) to write the equation of line r  

y - 8 = (-19/ 5) ( x - 4)

Simplifying, ti becomes

y = (-19/5) x + 76/5

For line s, we can use the points (-2, - 18) and ( 1, -3) to find the slope  ..

m = (y₂ - y₁) / (x₂ - x₁) = (-3 - (-18)) / (1 - (-2) ) = 5

Let's use the point (-2, -18) to write the equation of line s

y - (-18) = 5(x - (-2))

Simplifying we have

y = 5x - 8

Therefore, the equation of line s is y = 5x - 8.

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First, subtract 12 from 30, then multiply by 2. Therefore, option A is the correct answer.

The given expression is (30-12)×2.

Here, determine the difference between 30 and 12, that is 30-12=18

Multiply the difference by 2, that is

18×2=36

Therefore, option A is the correct answer.

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We can conclude that both probabilities in parts (a) and (b) are approximately 0.95, regardless of the sample size.

Assuming that the population is normally distributed with a mean of 100 and a standard deviation of 15, we can use the central limit theorem to approximate the sampling distribution of the sample means as normal, regardless of the sample size.

Since we're interested in the probability that the sample mean falls within a certain range, we can standardize the sampling distribution of the sample means using the formula:

z = ([tex]\bar X[/tex] - μ) / (σ / √n)

where [tex]\bar X[/tex] is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

For both parts (a) and (b), we want to find P(90 < [tex]\bar X[/tex] < 110), which we can rewrite as P((90 - 100) / (15 / √n) < ([tex]\bar X[/tex] - 100) / (15 / √n) < (110 - 100) / (15 / √n)).

Simplifying this expression, we get P(-2√n < z < 2√n), where z is a standard normal random variable.

The probability that a standard normal random variable falls within ±2 standard deviations of the mean is approximately 0.95. Therefore, we can conclude that both probabilities in parts (a) and (b) are approximately 0.95, regardless of the sample size.

This is because the sample size n only affects the standard deviation of the sampling distribution of the sample means (i.e., σ / √n), but not its shape or center. Therefore, the probability that the sample mean falls within a certain range is largely determined by the width of that range relative to the standard deviation of the sampling distribution, which is approximately constant regardless of the sample size.

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-2/pi is the average rate of change of y=cos(2x) on the interval [0, pi/2]. The correct answer is option a.

We can find the average rate of change of a function f(x) on an interval [a,b] using the formula:

average rate of change = (f(b) - f(a)) / (b - a)

Using this formula and the given function y = cos(2x) on the interval [0, pi/2], we get:

average rate of change = (cos(2(pi/2)) - cos(2(0))) / (pi/2 - 0)

= (-1 - 1) / (pi/2)

= -2/pi

Therefore, the answer is not one of the given options. The closest option is A.

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Complete Question

What is the average rate of change of y=cos(2x) on the interval [0, pi/2]?

A. -2/pi

B. -1

C. 0

D. Square root of 2/2

E. 4/pi

An exponential equation that best models this set of data is [tex]y = 6.003e^{0.1821x}[/tex]

In this scenario, the years would be plotted on the x-axis (x-coordinate) of the scatter plot while the town population would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.

On the Microsoft Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an exponential equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the years and town population, an exponential equation for the line of best fit is given by:

[tex]y = 6.003e^{0.1821x}[/tex]

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One share of Noshima Industries stock is worth $8.27 today if you require a 9% rate of return.

To calculate the present value of the future dividend payments, we can use the formula for the present value of a growing perpetuity:

PV = C / (r - g)

where PV is the present value, C is the annual cash flow (dividend), r is the required rate of return, and g is the annual growth rate of the cash flow.

In this case, the annual cash flow starts at $0.60 and then grows by 50% for two years, so we can calculate the cash flows for the next three years as follows:

Year 1: $0.60

Year 2: $0.60 × 1.5 = $0.90

Year 3: $0.90 × 1.5 = $1.35

After the third year, the dividend payments are expected to remain constant, so we can calculate the present value of perpetuity using the constant dividend payment of $1.35 per year.

Using a required rate of return of 9%, the annual growth rate for the first two years of 50%, and the constant growth rate of 0% after that, we can calculate the present value of the future cash flows as follows:

PV = 0.60/(0.09 - 0.50) + (0.90/(1+0.09)²)/(0.09 - 0.50) + (1.35/(1+0.09)³)/(0.09 - 0.00)

PV = $8.27174733765

PV = $8.27

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Note that the estimate of the number of females who are over 50 years of age based on the histogram are 7.

We can work out the frequency from the histogram by doing:

Frequency = Frequency Density  x Group Width

Now lets find the number of people above the age of 50, which is 2 area in the histogram.

Area 1 = 1.4 x 10  = 14

Area 2 = 0.7 x 30 = 21

Add to the total over 50

Total =

14 + 21 = 35

Recall that 20% are female so

20% of 35 = 7

Therefore,  the number of females who are over 50 years of age based on the histogram are 7.

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The work required to pump the water out of the spout is approximately 2026 ft-lb.

The work required to pump the water out of the spout depends on the weight of the water and the height of the spout.

The formula for work:

work = force x distance

The force required to lift the water is equal to its weight.

The weight of the water can be found using the density of water, which is 62.5 lb/ft³, and the volume of water in the tank.

Let's assume that the tank has a rectangular base of length L and width W, and a height H.

The volume of water in the tank is equal to the product of the length, width, and height of the tank:

volume = L x W x H

The weight of the water is equal to its volume times its density:

weight = volume x density

weight = L x W x H x 62.5

To pump the water out of the spout, we need to lift it to a height equal to the height of the water level in the tank.

Let's assume that the height of the water level in the tank is h.

The work required to pump the water out of the spout is then:

work = weight x h

work = L x W x H x 62.5 x h

Since we don't have values for L, W, H, and h, we can't calculate the exact work required.

The formula by assuming that the tank is a cube of side length s, and that the height of the water level in the tank is equal to the side length of the cube. In this case, we have:

volume = s³

weight = s³ x 62.5

work = s³ x 62.5 x s

work = 62.5 s⁴

To get the work required in ft-lb, we need to convert the weight from lb to ft-lb.

Since weight is force times gravity, and gravity is approximately 32.2 ft/s², we have:

weight in ft-lb = weight in lb x gravity

weight in ft-lb = s³ x 62.5 x 32.2

The work required in ft-lb by multiplying the weight in ft-lb by the height of the water level in the tank is also equal to the side length of the cube:

work in ft-lb = weight in ft-lb x s

work in ft-lb = s⁴ x 62.5 x 32.2

Plugging in s = 1 (since we don't have any value for the length of the side), we get:

work in ft-lb ≈ 2026

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The probability that David will select a blue Skittle from each bag is 17/30

Given that, David has 8 red Skittles, 6 blue Skittles, and 6 green Skittles in one bag and second bag contains 5 red Skittles, 4 blue Skittles, and 6 green Skittles.

He randomly selected 1 Skittle from each bag, we need to find the probability that David will select a blue Skittle from each bag,

So, the total number of skittles in one bag = 20

P(getting blue from one bag) = 6/20

The total number of skittles in 2nd bag = 15

P(getting blue from 2nd bag) = 4/15

P(selecting a blue Skittle from each bag) = 6/20 + 4/15 = 17/30

Hence, the probability that David will select a blue Skittle from each bag is 17/30

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The appropriate statistical test for a correlated groups experiment where the population standard deviation needs to be estimated to determine the significance of the sample results is the paired t-test.

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A quadrilateral with two pairs of opposite sides always parallel is a  parallelogram. A parallelogram is a quadrilateral with two pairs of opposite sides that are always parallel.

It is a special type of quadrilateral because its properties allow it to be manipulated in unique ways. For example, if one angle of a parallelogram is a right angle, it becomes a rectangle. If all four sides are congruent, it becomes a rhombus. If both pairs of opposite sides are congruent, it becomes a square.

If only one pair of opposite sides is congruent, it becomes a kite. And if it has two parallel sides that are not congruent, it becomes an isosceles trapezoid. A trapezoid also has two pairs of opposite sides, but they are not always parallel. Therefore, it cannot be the answer to this question. Overall, the parallelogram is a versatile shape with unique properties that make it an important concept in geometry.

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a. the proportion of Lime Cola bottles that have contents less than 755 mL is 0.7580. b. the proportion of Lime Cola bottles that have contents between 740 mL and 764 mL is 0.9936. c. the proportion of Lime Cola bottles that have contents more than 760 mL is 0.0446.

To solve this problem, we need to use the standard normal distribution with a mean of 0 and a standard deviation of 1. We can convert the given values to the standard normal distribution using the formula:

z = (x - μ) / σ

where z is the standard score, x is the given value, μ is the mean, and σ is the standard deviation.

(a) We need to find the proportion of Lime Cola bottles that have contents less than 755 mL. We can calculate the z-score as:

z = (755 - 751.5) / 5 = 0.7

Using the standard normal distribution table or calculator, we can find that the proportion of values less than 0.7 is approximately 0.7580. Therefore, the proportion of Lime Cola bottles that have contents less than 755 mL is 0.7580.

(b) We need to find the proportion of Lime Cola bottles that have contents between 740 mL and 764 mL. We can calculate the z-scores as:

z1 = (740 - 751.5) / 5 = -2.3

z2 = (764 - 751.5) / 5 = 2.5

Using the standard normal distribution table or calculator, we can find the proportion of values between -2.3 and 2.5 is approximately 0.9936. Therefore, the proportion of Lime Cola bottles that have contents between 740 mL and 764 mL is 0.9936.

(c) We need to find the proportion of Lime Cola bottles that have contents more than 760 mL. We can calculate the z-score as:

z = (760 - 751.5) / 5 = 1.7

Using the standard normal distribution table or calculator, we can find the proportion of values more than 1.7 is approximately 0.0446. Therefore, the proportion of Lime Cola bottles that have contents more than 760 mL is 0.0446.

(d) We need to find the proportion of Lime Cola bottles that have contents not between 746 mL and 754 mL. We can calculate the z-scores as:

z1 = (746 - 751.5) / 5 = -1.1

z2 = (754 - 751.5) / 5 = 0.5

Using the standard normal distribution table or calculator, we can find the proportion of values less than -1.1 and more than 0.5 is approximately 0.4088. Therefore, the proportion of Lime Cola bottles that have contents not between 746 mL and 754 mL is 0.4088.

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Answer:

7 < 7B < 14

Step-by-step explanation:

The equation 7 x B represents a product of two factors, where B is one of the factors. The equation states that the value of this product is between 7 and 14, exclusive. Therefore, B must be a fractional value between 1 and 2. Specifically, B could be any value between 1 and (2/7), exclusive, since any value greater than (2/7) would result in a product greater than 14, and any value less than 1 would result in a product less than 7, This will be represented as 7 < 7B < 14

Answer:

Step-by-step explanation:

A proportion is a fancy way of saying that two fractions are the same. To find out what the missing number is, we can do some magic called cross-multiplication, which means we multiply the numbers that are diagonal from each other and make them equal. For example, if we have a proportion like a/b = c/d, we can cross-multiply and get ad = bc. Then we can find the missing number by moving things around until it's by itself. Let's try this trick on this problem:

7/x+2 = 4/x+1

Cross-multiply:

7(x+1) = 4(x+2)

Spread out:

7x + 7 = 4x + 8

Take away 4x from both sides:

3x + 7 = 8

Take away 7 from both sides:

3x = 1

Cut both sides by 3:

x = 1/3

So, the answer is x = 1/3.

The correct statement is: the volume of the cylinder is 8 cubic inches more than that of the cone.

The volume of cone (V) = 1/3 * πr²h

Volume of cylinder (V) = πr²h

Where, r is the radius of their bases.

The base area formula for the cone = πr²

Calculate the area of the base of the cone that has a radius (r) of 2.5 in:

Area of the base of the cone = π(2.5)² ≈ 19.6

The volume of the cone (V) = 1/3 * πr²h = 1/3 * π * 2.5² * 6.5

≈ 42.5 cubic inches

Volume of cylinder (V) = πr²h = π * 2² * 4

≈ 50.3 cubic inches

The difference in volume = 50.3  - 42.5

= 7.8 ≈ 8 cubic inches

Therefore, the only statement that is true is: the volume of the cylinder is 8 cubic inches more than that of the cone.

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The rate at which the cost is changing with respect to g is 0.75 dollars per flat of geraniums. This means that for each additional flat of geraniums, Jane buys, her total cost increases by $0.75.

Given information:

Jane buys 12 flats of flowers.

Flats of geraniums cost $2. 00, and flats of roses cost $1. 25. The total cost, c, in terms of the number of flats of geraniums, g, is given by c=2. 00g+1. 25(12-g).

We can start by taking the derivative of the cost function, c, with respect to the number of flats of geraniums, g:

dc/dg = 2.00 - 1.25

Simplifying this expression gives:

dc/dg = 0.75

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Answer: To find the mass of the piece of mahogany wood, we need to use the formula:

mass = density x volume

Substituting the given values, we get:

mass = 0.71 g/cm³ x 1728 cm³

mass = 1228.88 g

Therefore, the mass of the piece of mahogany wood is 1228.88 g.

The amount of fencing needed to enclose the circular pig pin, which refers to its circumference, is approximately 57.8 ft.

The circumference of a circle describes its perimeter or the total distance around the circle.

There are two ways to compute the circumference using the diameter.

The circumference is equal to π × Diameter or 2πR

The diameter of the pin = 18.4 ft

Circumference formula = π × Diameter or 2πR, where π is the pi or 3.14, r is radius of ¹/₂ of diameter.

= 3.14 x 18.4

= 57.8 ft.

Thus, note that the Pi (π) is a special mathematical constant, showing the ratio of circumference to diameter of any circle.

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A cheeseburger contains 400 calories and a small order of fries contains 320 calories.

Let's denote the caloric content of a cheeseburger by "c" and the caloric content of a small order of fries by "f".

We can set up a system of equations based on the given information:

2c + f = 1120 (Equation 1)

3c + 2f = 1840 (Equation 2)

To solve for "c" and "f", we can use elimination or substitution. Let's use elimination:

Multiply Equation 1 by 2:

4c + 2f = 2240 (Equation 3)

Subtract Equation 3 from Equation 2:

3c + 2f - (4c + 2f) = 1840 - 2240

-c = -400

c = 400

Now we can substitute this value back into either Equation 1 or Equation 2 to solve for "f". Let's use Equation 1:

2(400) + f = 1120

800 + f = 1120

f = 320

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The value of the possible coordinates of point B is, (- 8, 10)

We have to given that;

The coordinates of point A are (4,1).

And, the x- coordinate of point B is -8.

Since, the distance between point A and B is 15 units.

Let us assume that,

the possible coordinates of point B is,

= (x, y)

= (- 8, y)

The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Hence, We get;

the distance between point A and B is,

⇒ d = √ (- 8 - 4)² + (y - 1)²

⇒ 15 = √144 + (y - 1)²

⇒ 225 = 144 + (y - 1)²

⇒ (y - 1)² = 225 - 144

⇒ (y - 1)² = 81

⇒ y - 1 = 9

⇒ y = 10

Thus, The value of the possible coordinates of point B is, (- 8, 10)

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Answer:4.333.

Step-by-step explanation:

To find the average rate of change over the interval [1,4], we can use the formula:

Average rate of change = (change in y)/(change in x)

In this case, the values of x and y are given in the table as follows:

x: 1 2 3 4

y: 8 14 17 21

So, the change in x over the interval [1,4] is 4 - 1 = 3, and the change in y is 21 - 8 = 13.

Plugging these values into the formula, we get:

Average rate of change = (13)/(3) = 4.333...

So, the average rate of change over the interval [1,4] is approximately 4.333.