Probability of a pair of dice landing on a 4 and on a 6

The y coordinate of the solution to this system of equations is -2

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

3x+y=-2 x-2y=4

Express properly

So, we have

3x + y = -2

x - 2y = 4

Multiply the second equation by -3

so, we have the following representation

3x + y = -2

-3x + 6y = -12

Add the equations to eliminate x

7y = -14

Divide both sides by 7

y = -2

Hence, the value of y is -2

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The equation sin(3x + 9) = cos(x + 5) has no solutions in the set of acute angles.

To find two acute angles that satisfy the equation sin(3x + 9) = cos(x + 5), we can use the trigonometric identity cos(x) = sin(π/2 - x) to rewrite the right-hand side of the equation as follows:

Dividing both sides of the equation by 4, we get:

x = -(1/2)πn - 1

So the solutions are given by:

x = -(1/2)π - 1 and x = -(3/2)π - 1

To check that these solutions make sense, we need to ensure that they are acute angles, i.e., angles that measure less than 90 degrees.

The first solution, x = -(1/2)π - 1, can be written in degrees as:

x ≈ -106.26 degrees

This angle is not acute, so it is not a valid solution.

The second solution, x = -(3/2)π - 1, can be written in degrees as:

x ≈ -286.87 degrees

This angle is also not acute, so it is not a valid solution.

Therefore, there are no acute angles that satisfy the equation sin(3x + 9) = cos(x + 5).

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The probability that a randomly selected student got an A in Math, but not English, is 8%

Let A be the event that a student got an A in Math, and B be the event that a student got an A in English. Then, we want to find P(A and not B), or the probability that a student got an A in Math, but not English.

We know that P(A and B) = 0.16, or the probability that a student got an A in both Math and English. We also know that P(B) = 0.37, or the probability that a student got an A in English. Therefore, the probability of a student getting an A in Math, given that they got an A in English, can be calculated using the formula for conditional probability:

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

P(A | B) = 0.16 / 0.37

P(A | B) = 0.43

This means that the probability of a student getting an A in Math, given that they got an A in English, is approximately 0.43.

To find the probability of a student getting an A in Math, but not English, we can subtract the probability of getting an A in both classes from the probability of getting an A in Math:

P(A and not B) = P(A) - P(A and B)

P(A and not B) = 0.24 - 0.16

P(A and not B) = 0.08

Therefore, the probability that a randomly selected student got an A in Math, but not English, is 0.08 or 8%.

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The time till you hit the water, given your height above the water, would be 0.76 seconds.

The maximum height above the pool you would get is 15.39 feet.

We are given h ( t ) = - 16 t ² + 5t + 15

You hit the water at 0 so the formula is:

0 = - 16 t ² + 5t + 15

Using the quadratic equation, we can solve:

t = ( - b ± √ ( b ² - 4  ac ) ) / 2a

t = ( - 5 ± √ ( 5 ² - 4 ( - 16 ) ( 15 ) )) / 2 (- 16)

t = (- 5 ± √ 985 ) / -32

t = 0. 76 seconds

The vertex of the parabolic function would be:

= - b / 2a

= - 5 / ( 2 x - 16 )

=  0. 15625 seconds

Maximum height is therefore:

= -16 ( 0. 15625 ) ² + 5 ( 0. 15625 ) + 15

=  15.39 feet

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The standard deviation of the sampling distribution is 2.53. It would not be appropriate to use a normal distribution to model the sampling distribution.

(a) When dealing with a sampling distribution, the mean of the sampling distribution (μ_x) is equal to the mean of the population (μ). In this case, the mean of the population is 65. Therefore, the mean of the sampling distribution is also 65.

To calculate the standard deviation of the sampling distribution (σ_x), you will use the following formula: σ_x = σ / √n, where σ is the standard deviation of the population and n is the sample size. In this case, the standard deviation of the population is 8 and the sample size is 10. So the standard deviation of the sampling distribution is:

σ_x = 8 / √10 ≈ 2.53

(b) The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided that the population is not heavily skewed or has extreme outliers. Since the scores are extremely skewed left and the sample size is only 10, it would not be appropriate to use a normal distribution to model the sampling distribution. A larger sample size would be needed to use a normal distribution model.

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The quantity that gives maximum profit is approximately 111.55 units.

To find the quantity that gives maximum profit, we need to first find the revenue function and then the profit function.

The revenue function is given by:

R(x) = xp = x(120,000 - 150x)

The profit function is given by:

P(x) = R(x) - C(x) = x(120,000 - 150x) - (700/x + 300x + x²)

To find the quantity that gives maximum profit, we need to find the derivative of the profit function and set it equal to zero:

P'(x) = 120,000 - 300x - 700/x² - 2x

Setting P'(x) equal to zero and solving for x, we get:

120,000 - 300x - 700/x² - 2x = 0

Multiplying both sides by x^2, we get:

120,000x² - 300x³ - 700 - 2x³ = 0

Simplifying, we get:

300x³ + 2x³ - 120,000x² - 700 = 0

Dividing both sides by 2, we get:

151x³ - 60,000x² - 350 = 0

Using a graphing calculator or numerical methods, we can find that the real root of this equation is approximately x = 111.55.

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The parabola with a focus of (-4, 5) and a directrix of y = -9 is vertically oriented, opens upward, has a vertex at (-4, -2), and its equation is (x + 4)^2 = 28(y + 2).

1. The parabola is vertically oriented since the directrix is a horizontal line.
2. The vertex of the parabola is equidistant from the focus and the directrix. To find the vertex, we can calculate the midpoint between the focus and a point on the directrix with the same x-coordinate: (-4, -9 + (5 - (-9))/2) = (-4, -9 + 7) = (-4, -2).
3. The parabola opens upward because the focus is above the directrix.
4. The equation of the parabola can be found using the vertex form: (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance between the vertex and the focus or directrix. In this case, (h, k) = (-4, -2), and p = (5 - (-2)) = 7. The equation is therefore (x + 4)^2 = 28(y + 2).

In summary, the parabola with a focus of (-4, 5) and a directrix of y = -9 is vertically oriented, opens upward, has a vertex at (-4, -2), and its equation is (x + 4)^2 = 28(y + 2).

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The measure of the central angle that intercepts the arc of the circle is approximately 286.48 degrees.

A circle's circumference is comprised of arcs. In other words, if you draw any two locations on a circle's circumference, the curved line that joins them around the border of the circle is referred to as an arc.

The formula for the relationship between arc length and central angle is:

arc length = radius x central angle

We are given the arc length and radius, so we can solve for the central angle:

10 = 2 x central angle

Dividing both sides by 2, we get:

central angle = 10/2

= 5 radians

To convert from radians to degrees, we multiply by 180/π:

central angle = 5 x 180/π

≈ 286.48 degrees.

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The given inequality is 3x - 12 ≤ -18. To solve for x, we can add 12 to both sides of the inequality to obtain 3x ≤ -6. Then, dividing both sides of the inequality by 3 gives x ≤ -2. Therefore, any value of x less than or equal to -2 will satisfy the inequality.

In solving the inequality, we first used the addition property of inequalities to add 12 to both sides of the inequality. This property states that if a < b, then a + c < b + c, where c is any real number. By adding 12 to both sides, we were able to isolate the variable term on one side of the inequality.

Next, we used the division property of inequalities to divide both sides of the inequality by 3. This property states that if a < b and c > 0, then a/c < b/c. By dividing both sides of the inequality by 3, we were able to solve for x.

Finally, we found that any value of x less than or equal to -2 will satisfy the inequality. This means that the solution set for the inequality is {x | x ≤ -2}. We also verified that x = -2 is a valid solution to the inequality, which confirms our solution.

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The measure of the angle ∠S, obtained using the law of cosines is about 59.4°

The law of cosines states that the square of the length of a side of a triangle is equivalent to the sum of the squares of the other two sides less the product of the length of the other two sides and the angle between them. Mathematically; a² = b² + c² - 2·b·c·cos(A)

Where;

a, b, and c = The length of the sides of the triangle

A = The angle between b and c

The lengths of the sides of the triangle, obtained from a similar triangle are;

s = 50 inches, t = 58 inches, and u = 27 inches

The measure of the angle ∠S is required

The angle ∠S is the angle facing TU or the side s

According to law of cosines, we get;

s² = t² + u² - 2·t·u·cos(∠S)

Therefore;

50² = 58² + 27² - 2 × 58 × 27 × cos(∠S)

2 × 58 × 27 × cos(∠S) = 58² + 27² - 50²

cos(∠S) = (58² + 27² - 50²)/(2 × 58 × 27)

∠S = cos((58² + 27² - 50²)/(2 × 58 × 27)) ≈ 59.4°

The measure of the angle ∠S is about 59.4°

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So, the answer is:
(a) f(x) does not exist at xa = 16.
(b) lim f(x) as x approaches 16 does not exist.
(c) None of the conditions for continuity are met at xa = 16.

To find the value of xa where the function is discontinuous, we need to look for any points where the function is undefined or where the left and right limits of the function are not equal.
(a) From the given information, we know that the function is undefined at xa = 16. So, this is the point of discontinuity.
(b) To find the left and right limits at xa = 16, we need to approach the point from both sides of the function. So,
lim f(x) as x approaches 16 from the left (denoted as lim-) = Im = 0
lim f(x) as x approaches 16 from the right (denoted as lim+) = Im = 16
Since the left and right limits are not equal, the limit as x approaches 16 does not exist. So,
lim f(x) as x approaches 16 (denoted as lim) does not exist.
(c) To determine which conditions for continuity are not met, we need to check if the function satisfies the three conditions for continuity at xa = 16.
i) The function must be defined at xa = 16. Since the function is undefined at xa = 16, this condition is not met.
ii) The left and right limits of the function must exist and be equal at xa = 16. Since the left and right limits are not equal, this condition is not met.
iii) The value of the function at xa = 16 must be equal to the limit of the function at xa = 16. Since the limit does not exist, this condition is also not met.
Therefore, none of the conditions for continuity are met at xa = 16.
So, the answer is:
(a) f(x) does not exist at xa = 16.
(b) lim f(x) as x approaches 16 does not exist.
(c) None of the conditions for continuity are met at xa = 16.
Note: The terms "discontinuous" and "continuity" are used throughout the explanation to describe the concept and the point of interest. The term "function" refers to the given function that we are analyzing.

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WHI Health Fund pays the most commission on the purchase of 100 shares with a commission of $96.00. Thus, option d is correct.

Funds offer price for  Upton Group = $18.96 - $18.47

Funds offer price for Green Energy fund = $18.01 - $17.29

Funds offer price for TJH Small-Cap fund = $19.05 - $18.43

Funds offer price for WHI Health fund = $20.96 - $20.00

To calculate the commission on purchasing shares, we need to find the allowance between the price ranges and then multiply the value by 100.

For the Upton Group fund, Commission = (Offer price - NAV) * 100

= ($18.96 - $18.47) * 100

= $49.00

For the Green Energy fund, Commission = (Offer price - NAV) * 100

= ($18.01 - $17.29) * 100

= $72.00

For the TJH Small-Cap fund, Commission = (Offer price - NAV) * 100

= ($19.05 - $18.43) * 100

= $62.00

For the WHI Health fund, Commission = (Offer price - NAV) * 100

= ($20.96 - $20.00) * 100

= $96.00

Therefore, we can conclude that the WHI Health fund pays the most commission of $96.00.

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The y-intercept of the line is -13.

To find the y-intercept of the line, we can use the slope-intercept form of the equation of a line: y = mx + b,

where m is the slope and b is the y-intercept.

Given that the line passes through the point (-10, 52) and has a slope of -6.5, we can substitute these values into the equation:

52 = -6.5(-10) + b

Simplifying the equation:

52 = 65 + b

To isolate b, we subtract 65 from both sides:

52 - 65 = b

Simplifying further:

b = -13

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Answer:

A= 254.47 in

C= 56.55 in^2

Step-by-step explanation:

formula for area is πr^2 (radius is r)

circumference formula is πd or 2πr (diameter is d, radius is r)

I don't know what does C and A means but if A means area and C means circumference,

C = 56.52in

A = 254.34

Given the scenerio in the picture about corn, the statement that is true looking at a significance level of 5% is The result is not statistically significant which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate.

Looking at the statement "The result is not statistically significant,"this means that the p-value (probability value) of the test was greater than 0.05. It could have 0.15 oe 0.2.

When a test is greater than 0.05 or 5 % significance level, it shows that the what is happening to the corn (increase in growth rate) could have been as a result of chance alone.

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Based on the above, Keenan paid $9,420 in federal income tax in 2009.

Let X be the amount Keenan paid in taxes in 2009.

According to the problem, Keenan paid 70% less in 2010 than he did in 2009. This means that he paid only 30% of what he paid in 2009, since 100% - 70% = 30%.

We can express this mathematically as:

0.30X = 2,826

To solve for X, we can divide both sides of the equation by 0.30:

X = 2,826 ÷  0.30

X = 9,420

Therefore, Keenan paid $9,420 in federal income tax in 2009.

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Answer:

let x be the no.

So, 5 less than the square of a number in an algebraic expression is:

x^2 - 5

Answer:

6 30/31 as fraction and with Long Division its 6 with 30 Remainder

The radius of the circular fountain is approximately 17 feet.

The formula for the volume of a circular fountain is given by V = πr^2h, where V is the volume, r is the radius, and h is the depth. In this case, we are given that the depth of the fountain is 3 feet and the volume is 1800 cubic feet. So we can plug in these values into the formula and solve for r as follows:

1800 = πr^2(3)

Simplifying this equation, we get:

r^2 = 600/π

Taking the square root of both sides, we get:

r = sqrt(600/π)

Using a calculator to approximate the value of sqrt(600/π), we get:

r ≈ 17

Therefore, the radius of the circular fountain is approximately 17 feet when rounded to the nearest whole number.

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Ms. Summers gives each child 1/16 gallon of milk.

The amount of milk that Ms. Summers gives to each child can be represented by the following expression:

(1/4 gallon of milk - 1/8 gallon of milk) / 2

This expression represents the amount of milk that remains after Ms. Summers drinks 1/8 gallon of milk, divided equally between her two children.

Simplifying the expression, we get:

(2/8 - 1/8) / 2 = 1/8 / 2 = 1/16

Therefore, Ms. Summers gives each child 1/16 gallon of milk.

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Using the given information, Riley bought 3.36 pounds of chicken thighs

From the question, we are to calculate the number of pounds of chicken thighs that Riley bought

We can use proportionality to find how many pounds of chicken thighs Riley bought.

If 5 pounds of chicken thighs cost $23.75, then we can write the following proportion:

Cost/Weight = $23.75/5 lb

We can use this proportion to find the cost per pound of chicken thighs:

That is,

Cost/Weight = $23.75/5 lb = $4.75/lb

Now we can use this rate to find how many pounds of chicken thighs Riley bought:

Cost of chicken thighs bought = $15.96

Weight of chicken thighs = Cost of chicken thighs / Cost per pound of chicken thighs

Weight of chicken thighs bought = $15.96 / $4.75/lb ≈ 3.36 lb

Hence, Riley bought 3.36 pounds of chicken thighs.

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After solving equation, Katya needs 5 or 6 display cases to display all of her badges, depending on how she chooses to group them.

The number of badges Katya has is not explicitly given in the problem. Therefore, we need to first find the total number of badges Katya has earned by subtracting the number of parks for which she doesn't have badges from the total number of participating parks.

The equation to use for this is:

b = 207 - 165

where b is the total number of badges Katya has earned.

Option A is the correct equation, which simplifies to:

b = 42

This means Katya has earned 42 badges.

Next, we need to find out how many display cases Katya needs. We know that she can fit 8 badges in one case, so we can use division to find the number of cases needed:

c = b / 8

where c is the number of cases needed.

Substituting the value of b from the previous equation, we get:

c = 42 / 8

Option C is the correct equation, which simplifies to:

c = 5.25

This means that Katya needs 5 or 6 display cases to display all of her badges, depending on how she chooses to group them.

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Management should hire 25 additional vendors to maximize the number of sodas sold.

Let x be the number of additional vendors that management hires. Then the total number of vendors is 40 + x, and the yield per vendor is 200 - 4x (since the yield decreases by 4 for each additional vendor).

The total number of sodas sold is the product of the number of vendors and the yield per vendor:

Total sodas sold = (40 + x) * (200 - 4x)

To maximize the number of sodas sold, we take the derivative of this expression with respect to x and set it equal to zero:

d/dx [(40 + x) * (200 - 4x)] =

Expanding and simplifying, we get:

-8x² + 120x + 8000 = 0

Dividing both sides by -8, we get:

x² - 15x - 1000 = 0

Using the quadratic formula, we solve for x:

x = (15 ± sqrt(15² + 411000)) / 2

x = (15 ± 35) / 2

x = -10 or x = 25

Since we can't hire a negative number of vendors, the only sensible solution is x = 25. Therefore, management should hire 25 additional vendors to maximize the number of sodas sold.

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The probability of Dai ordering a regular, medium milk is 1/18.

There are 2 types of milk (regular and chocolate), and 3 sizes (small, medium, and large), and 3 levels of fat content (skim, 2%, and whole). So the total number of possible milk orders is:

2 (types of milk) x 3 (sizes) x 3 (fat content) = 18

Dai needs to choose regular milk and medium size, so there is only one way she can order this combination.

The probability of Dai ordering a regular, medium milk is the number of ways she can order a regular, medium milk divided by the total number of possible milk orders:

1 (number of ways to order a regular, medium milk) / 18 (total number of possible milk orders) = 1/18

So the probability that Dai orders a regular, medium milk is 1/18 or approximately 0.056 (rounded to three decimal places).

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The secant of a °45 angle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. In this case, the adjacent side is also a leg of length 5, so:

secant(45) = hypotenuse/adjacent = 5√2/5 = √2

The secant function, denoted as sec(x), is a trigonometric function that is defined as the reciprocal of the cosine function, cos(x).

In other words,

sec(x) = 1 / cos(x)

The secant function is defined for all values of x except for those where the cosine function is equal to zero, which corresponds to the values x = (2n+1)π/2 where n is an integer. At these points, the secant function is undefined.

In a °45-°45-°90 triangle, the two legs are congruent, so if the length of the hypotenuse is 5√2, then each leg has a length of:

leg = hypotenuse/√2 = (5√2)/√2 = 5

The sine of a °45 angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the angle is a leg of length 5, so:

sine(45) = opposite/hypotenuse = 5/5√2 = 1/√2 = √2/2

The secant of a °45 angle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. In this case, the adjacent side is also a leg of length 5, so:

secant(45) = hypotenuse/adjacent = 5√2/5 = √2

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To use the Picard-Lindelöf iteration to find a sequence of approximate solutions to the initial value problem y'(t) = 5y(t) + 1, y(0) = 0, we start with the initial approximation y_0(t) = 0. Then, for each n ≥ 0, we define y_{n+1}(t) to be the solution to the initial value problem y'(t) = 5y_n(t) + 1, y_n(0) = 0. In other words, we plug the previous approximation y_n into the right-hand side of the differential equation and solve for y_{n+1}.Using this procedure, we can find the first few elements of the sequence {y_n} as follows:y_0(t) = 0y_1(t) = ∫ (5y_0(t) + 1) dt = ∫ 1 dt = ty_2(t) = ∫ (5y_1(t) + 1) dt = ∫ (5t + 1) dt = (5/2)t^2 + ty_3(t) = ∫ (5y_2(t) + 1) dt = ∫ (5(5/2)t^2 + 5t + 1) dt = (25/6)t^3 + (5/2)t^2 + tTherefore, the first few elements of the sequence {y_n} are y_0(t) = 0, y_1(t) = t, y_2(t) = (5/2)t^2 + t, and y_3(t) = (25/6)t^3 + (5/2)t^2 + t.

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To use the Picard-Lindelöf iteration method to find the first few elements of a sequence {y_n} of approximate solutions to the initial value problem y'(t) = 5y(t) + 1, y(0) = 0, we first set up the integral equation for the iteration:

y_n+1(t) = y(0) + ∫[5y_n(s) + 1] ds from 0 to t

Since y(0) = 0, the equation becomes:

y_n+1(t) = ∫[5y_n(s) + 1] ds from 0 to t

Now, let's calculate the first few approximations:

1. For n = 0, we start with y_0(t) = 0:
y_1(t) = ∫[5(0) + 1] ds from 0 to t = ∫1 ds from 0 to t = s evaluated from 0 to t = t

2. For n = 1, use y_1(t) = t:
y_2(t) = ∫[5t + 1] ds from 0 to t = 5/2 s^2 + s evaluated from 0 to t = 5/2 t^2 + t

3. For n = 2, use y_2(t) = 5/2 t^2 + t:
y_3(t) = ∫[5(5/2 t^2 + t) + 1] ds from 0 to t = ∫(25/2 t^2 + 5t + 1) ds from 0 to t = 25/6 t^3 + 5/2 t^2 + t

These are the first few elements of the sequence {y_n} of approximate solutions to the initial value problem using the Picard-Lindelöf iteration method.

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The cost of a small cup of coffee is $2.97, and the cost of a piece of banana bread is $5.13.

Let x represent the cost of a small coffee and y represent the cost of a piece of banana bread. We know:

Cost of muffin: $3.51

y = x + $2.16

5(x + $3.51) + 2(x + y) = $48.60

Substitute y with x + $2.16:

5(x + $3.51) + 2(x + (x + $2.16)) = $48.60

Solve for x:

9x + $21.87 = $48.60

9x = $26.73

x = $2.97

Find y:

y = x + $2.16

y = $2.97 + $2.16

y = $5.13

A slice of banana bread costs $5.13, while a small coffee costs $2.97.

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The variance of the given data set is 18603.39.

To find the variance of the given data set {4, 44, 404, 244, 4, 74, 84, 64}, follow these steps:

Step 1: First, we need to find the mean of the data set:

Mean = (4 + 44 + 404 + 244 + 4 + 74 + 84 + 64) / 8 = 120.5

Step 2: Next, we calculate the deviation of each data point from the mean:

(4 - 120.5) = -116.5

(44 - 120.5) = -76.5

(404 - 120.5) = 283.5

(244 - 120.5) = 123.5

(4 - 120.5) = -116.5

(74 - 120.5) = -46.5

(84 - 120.5) = -36.5

(64 - 120.5) = -56.5

Step 3: Now, we square each deviation:

[tex](-116.5)^2 = 13556.25\\(-76.5)^2 = 5852.25\\(283.5)^2 = 80322.25\\(123.5)^2 = 15252.25\\(-116.5)^2 = 13556.25\\(-46.5)^2 = 2162.25 \\(-36.5)^2 = 1332.25\\(-56.5)^2 = 3192.25[/tex](-116.5)^2 = 13556.25

Step 4: We add up all the squared deviations:

13556.25 + 5852.25 + 80322.25 + 15252.25 + 13556.25 + 2162.25 + 1332.25 + 3192.25 = 130223.75

Step 5: We divide the sum of the squared deviations by the number of data points minus 1 to get the variance:

Variance = 130223.75 / 7 = 18603.39 (rounded to two decimal places)

Therefore, the variance of the data set is 18603.39.

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Answer:

k = 1

Step-by-step explanation:

We can use the table for g(x) = f(kx) to find the value of k.

Notice that when x = -2, we have g(-2) = f(k(-2)) = f(-2k) = 64. Similarly, when x = 2, we have g(2) = f(k(2)) = f(2k) = 64.

Using the fact that f(x) is a quadratic function, we can see that its axis of symmetry passes through the vertex at (0, 0), which means that the x-coordinate of the vertex is 0. This tells us that the coefficient of the x term in f(x) is 0, so the function can be written as f(x) = ax^2 + bx + c, where a is not equal to 0.

Using the points (−2,4), (−1,1), (0,0), (1,1), and (2,4), we can write a system of equations to solve for a, b, and c:

a(-2)^2 + b(-2) + c = 4

a(-1)^2 + b(-1) + c = 1

a(0)^2 + b(0) + c = 0

a(1)^2 + b(1) + c = 1

a(2)^2 + b(2) + c = 4

Simplifying and rearranging, we get:

4a - 2b + c = 4

a - b + c = 1

c = 0

a + b + c = 1

4a + 2b + c = 4

Substituting c = 0 into the system, we get:

4a - 2b = 4

a - b = 1

a + b = 1

4a + 2b = 4

Solving this system of equations, we get a = 1, b = -1, and c = 0.

Substituting these values into g(x) = f(kx), we get:

g(x) = f(kx) = x^2 - x

Substituting the values from the table into this equation, we get:

g(-2) = 4 = (-2)^2 - (-2) = 4k

g(2) = 4 = (2)^2 - (2) = 4k

Solving for k, we get k = 1 or k = -1/4.

However, we need to check which value of k satisfies all the points in between -2 and 2, so we can check g(-1) = 1 = (-1)^2 - (-1) = k, and g(1) = 1 = (1)^2 - (1) = k.

Thus, the value of k that satisfies all the points is k = 1, and therefore the answer is:

k = 1

We can use the information given to find the value of k.

Since the vertex of the quadratic function f(x) is at (0,0), we know that the equation for f(x) is in the form of f(x) = ax^2 for some constant a.

Using the point (-2, 4) on the graph of f(x), we can set up the equation 4 = 4a, which gives us a = 1.

So, the equation for f(x) is f(x) = x^2.

Now, we can use the table for g(x) = f(kx) to find the value of k.

When x = -2, we have g(-2) = f(k(-2)) = f(-2k) = 4k^2.

Similarly, when x = 2, we have g(2) = f(k(2)) = f(2k) = 4k^2.

We also know that g(0) = f(k(0)) = f(0) = 0, and g(-1) = f(k(-1)) = f(-k) = k^2 and g(1) = f(k(1)) = f(k) = k^2.

Using the values from the table, we can set up the following system of equations:

4k^2 = 64

k^2 = 16

0 = 0

k^2 = 16

The only solution that works for all of these equations is k = 4 or k = -4.

Therefore, the value of k is either k = 4 or k = -4.

Based on the Root Test, the series Σ^n√14 is convergent.

Hi! To determine if the series Σ^n√14 is convergent or divergent, we need to analyze the terms involved. The series can be written as:

Σ (n√14)

This is a sum of terms, where each term is the n-th root of 14, and we want to find out if the sum converges or diverges as n goes to infinity.

In this case, the series is a type of p-series, where the terms follow the general form of 1/n^p. To be a convergent p-series, p must be greater than 1. Here, the terms are in the form of 14^(1/n), which can be rewritten as (14^(1))^(-n) or 14^(-n). This is not a p-series, as the exponent is not in the form of 1/n^p.

To further analyze the series, we can use the Divergence Test. If the limit of the terms as n goes to infinity is not equal to zero, then the series is divergent. So, let's find the limit:

lim (n → ∞) (14^(-n))

As n approaches infinity, the exponent -n becomes increasingly negative, and 14^(-n) approaches 0. However, the Divergence Test is inconclusive in this case, as it only confirms divergence if the limit is not equal to zero.

To determine convergence or divergence, we can use the Root Test. The Root Test states that if the limit of the n-th root of the absolute value of the terms as n goes to infinity is less than 1, then the series converges. Let's find the limit:

lim (n → ∞) |(14^(-n))|^(1/n)

This simplifies to:

lim (n → ∞) 14^(-1)

Since 14^(-1) is a constant value less than 1, the limit is less than 1.

Thus, based on the Root Test, the series Σ^n√14 is convergent.

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