A researcher believes there is a difference in the mean number of days before visible results begin to show among three types of facial creams that reduce wrinkle lines. Several consumers are randomly selected and given one of the three creams. Each participant then recorded the number of days it took to see results. The results are shown in the table. Based on these data, can you conclude that there is a difference between the mean number of days for these three creams? Use a 0.025 level of significance and assume the population distributions are approximately normal with equal population variances. Cream #1 Cream #2 Cream #3 12 17 15 19 16 14 15 16 16 15 17 11 16 a. We reject the null hypothesis and conclude that there is sufficient evidence at a 0.025 level of significance to support the claim that the mean number of days before visible results begin to show is different between the three types of facial creams. b. We fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.025 level of significance to support the claim that the mean number of days before visible results begin to show is different between the three types of facial creams. c. We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.025 level of significance to support the claim that the mean number of days before visible results begin to show is different between the three types of facial creams. d. We reject the null hypothesis and conclude that there is insufficient evidence at a 0.025 level of significance to support the claim that the mean number of days before visible results begin to show is different between the three types of facial creams.

Answer:3 1/2 times 5= 15  1/10 - 7 1/4 = 8 6/40 or 8 3/20

Step-by-step explanation:

Answer:

3/4 or 0.75 tons

Step-by-step explanation:

The sum of the two months is 7 1/4 which is 29/4. For the first month, the company used 3 1/2 (7/2 or 14/4)

(29-14)/4=15/4.

Thus the second month you use 15/4.

Don’t simplify it yet-
15/4 divided by 5 is 3/4.

Boolean algebra is a type of algebra that deals with binary variables and logical operations. In this case, we're simplifying expressions using Boolean algebra.

1. ABC +(A+B+7)

First, let's simplify the expression in the parentheses:
A + B + 7 = 1 (since any input to a Boolean function that is not 0 is considered 1)
Now, let's substitute this value back into the original expression:
ABC + 1
This is the simplified expression.

2. (A+Ā)(AB+ ABC)

Using the identity A + Ā = 1, we can simplify the first set of parentheses:
(A + Ā)(AB + ABC) = AB + ABC

3. (B+BC)(B+ BC)(B+D)

Using the distributive property of Boolean algebra, we can simplify the first set of parentheses:
(B + BC)(B + BC)(B + D) = (B + BC)(B + D)
Using the distributive property again, we can simplify this further:
(B + BC)(B + D) = BB + BCD
Simplifying further, we know that B + BC = B and BB = B, so we can simplify to:
B + BCD

So, the simplified expressions are:
1. ABC + 1
2. AB + ABC
3. B + BCD

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The expanded form of the series [tex]\sum_{n=6}^{13}(2)^{n -1}[/tex] is given by  (2⁵) + (2⁶) + (2⁷) + (2⁸) + (2⁹) + (2¹⁰) + (2¹¹) + (2¹²) and the sum of the given series is equal to 8160.

The series is equal to,

[tex]\sum_{n=6}^{13}(2)^{n -1}[/tex]

First expand the series by plugging in the values of n from 6 to 13,

= 2⁶⁻¹ + 2⁷⁻¹ + 2⁸⁻¹ + 2⁹⁻¹ + 2¹⁰⁻¹ + 2¹¹⁻¹ + 2¹²⁻¹ + 2¹³⁻¹

= (2⁵) + (2⁶) + (2⁷) + (2⁸) + (2⁹) + (2¹⁰) + (2¹¹) + (2¹²)

Now,  use the formula for the sum of a geometric series to find the sum of this series,

S = a(1 - rⁿ)/(1 - r)

Here, a is the first term of the series,

r is the common ratio which is equals to 2 ,

and n is the number of terms in the series = 8.

Using this formula,  find the sum of the series we have,

S = (2⁵)(1 - 2⁸)/(1 - 2)

  = 32( 1-256 ) / (-1)

  = 8160

Therefore, the expanded form of the series is equal to (2⁵) + (2⁶) + (2⁷) + (2⁸) + (2⁹) + (2¹⁰) + (2¹¹) + (2¹²) and the sum of the series is 8160.

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The above question is incomplete, the complete question is:

What are the expanded form and sum of the series

[tex]\sum_{n=6}^{13}(2)^{n -1}[/tex]

The dimensions of the new pyramid is 8 times the original volume.

The area of the pyramid's base and its height are exactly proportional to its volume, hence the volume of any pyramid is equal to the area of the base times the height of the pyramid divided by three.

Knowing the formula of the pyramid is:

1/3 x a x b x h

If the dimensions are doubled, it will be :

1/3 x 2a x 2b x 2h

So:

v2= 1/3 x 2a x 2b x 2h

v2 = 1/3 x 8 x a x b x h

Hence, The new volume is 8 times more than the original.

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Where the logistic growth function is f( t) = 116000/(1+5200e⁻t)

a) 22 people were initially affected

b) after 4 weeks the numbers increased to 1,205 approximately

c) the limiting size of the population that becomes ill is 116,000.

The logistic equation (also known as the Verhulst model or logistic growth curve) is a population growth model developed by Pierre Verhulst (1845, 1847).

a) since when the epidemic began, the initial number of infection will always be zero, thus:

f( t) = 116000/(1+5200e⁻t )

Note that e is the mathematical constant known as Euler's number or the natural base. e = 2.71828

f(0) = 116000/(1+5200e⁻0 )

f(0) = 116000/(1+5200 (2.71828)⁻0 )

f(0) = 116000/(1+5200 (1) )

f(0) = 116000/(1+5200 )

f(0) = 116000/(5201 )

f(0) = 22.3034031917

f (0) [tex]\approx[/tex] 22

B) by the fourth week,

the expression became f(4) = 116000/(1+5200e⁻4 )

f (4) = 116000 /(1+5200 (e)⁻4 )
f (4) = 1205.30347384
f (4)  [tex]\approx[/tex] 1205

C)

Since the logistic curve is given as....

f(t) = 116000/(1+5200e⁻t )

as t becomes smaller and smaller nearing 0, the denominator will be almost 1

So
f(t) = 116000/(1+5200e⁻ ∞ )

f(t) = 116000/(1+0 )

f(t) = 116,000

As you can see, the limit to the size of the population that can fall ill is 116,000 peple.

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Answer:

2nd and 4th

Step-by-step explanation:

In the second quadrant, the x-coordinate is negative and the y-coordinate is positive. If we let the x-coordinate be -a, then the y-coordinate will be a. Therefore, the point will have the form (-a, a), and the y-coordinate will be the opposite of the x-coordinate.

In the fourth quadrant, the x-coordinate is positive and the y-coordinate is negative. If we let the x-coordinate be a, then the y-coordinate will be -a. Therefore, the point will have the form (a, -a), and again, the y-coordinate will be the opposite of the x-coordinate.

To design the optimal contract that induces both workers to exert effort all the time, consider the following steps:

1. Determine the joint probabilities of success for each combination of efforts:
  - Both workers shirk: Probability of success is p0.
  - One worker shirks and the other works hard: Probability of success is p1.
  - Both workers work hard: Probability of success is p2.

2. Identify the complementarity or substitutability of workers' efforts:
  - Since p2 > p1 > p0, the workers' efforts are complementary. This means that the success probability increases when both workers exert effort, as compared to only one worker doing so.

3. Design the optimal contract based on complementarity:
  - The principal should offer a contract with a bonus B, paid only if the project is successful.
  - To incentivize both workers to exert effort, the bonus should satisfy the following condition:
    B > 2c / (p2 - p1)

This ensures that the benefit of exerting effort (i.e., receiving the bonus) outweighs the cost of effort (c) for both workers. Since the workers' efforts complement each other, they will be more likely to exert effort knowing that their combined efforts increase the probability of project success and receiving the bonus.

In summary, the optimal contract should offer a bonus B that satisfies B > 2c / (p2 - p1) and is paid only upon project success. This contract incentivizes both workers to exert effort all the time, as their efforts complement each other and increase the probability of project success.

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The ratio StartFraction 14 Over 35 EndFraction is equivalent to the other three ratios, and the ratio that is different from the others is StartFraction 8 Over 20 EndFraction.

To determine which ratio is not equivalent to the others, we need to simplify each ratio to its lowest terms.

Give the following ratios :

[tex]2 / 5 = 6 /10 = 8 / 20 = 12 / 30.[/tex]

- StartFraction 2 Over 5 EndFraction: This ratio is already in its simplest form.

- StartFraction 6 Over 10 EndFraction: We can simplify this ratio by dividing both the numerator and denominator by their greatest common factor (GCF), which is 2.

-StartFraction 6 Over 10 EndFraction = StartFraction 3 Over 5 EndFraction

- StartFraction 8 Over 20 EndFraction: We can simplify this ratio by dividing both the numerator and denominator by their GCF, which is 4.

StartFraction 8 Over 20 EndFraction = StartFraction 2 Over 5 EndFraction

- StartFraction 12 Over 30 EndFraction: We can simplify this ratio by dividing both the numerator and denominator by their GCF, which is 6.

StartFraction 12 Over 30 EndFraction = StartFraction 2 Over 5 EndFraction

Therefore, the ratios StartFraction 6 Over 10 EndFraction, StartFraction 8 Over 20 EndFraction, and StartFraction 12 Over 30 EndFraction are all equivalent to StartFraction 2 Over 5 EndFraction. The ratio that is different from the others is StartFraction 14 Over 35 EndFraction, which can be simplified by dividing both the numerator and denominator by their GCF, which is 7.

[tex]StartFraction 14 Over 35 EndFraction = StartFraction 2 Over 5 EndFraction.[/tex]

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The appropriate alternative hypothesis is: Hₐ: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] does not equal 0, where [tex]\mu_{sc[/tex] is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme, and [tex]\mu_{dc[/tex] is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

An assertion used in statistical inference experiments is known as the alternative hypothesis. It is indicated by Hₐ or H₁ and runs counter to the null hypothesis.

The appropriate alternative hypothesis is: Hₐ: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] does not equal 0, where [tex]\mu_{sc[/tex] is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme, and [tex]\mu_{dc[/tex] is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

This hypothesis is appropriate because it is testing whether there is a statistically significant difference in the mean time to complete all matches between the two matching schemes. The null hypothesis would be that there is no difference in the mean time between the two schemes, i.e., H₀: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] = 0.

To test this hypothesis, we can use a paired t-test, which compares the mean difference between the two sets of measurements (in this case, the time to complete all matches using the two matching schemes) to the standard error of the mean difference. If the t-test results in a p-value that is smaller than the chosen significance level (typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the mean times for the two matching schemes.

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The appropriate alternative hypothesis is: Hₐ:  - does not equal 0, where  is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme and  is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

An assertion used in statistical inference experiments is known as the alternative hypothesis. It is indicated by Hₐ or H₁ and runs counter to the null hypothesis.

The appropriate alternative hypothesis is: Hₐ:  - does not equal 0, where  is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme and is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

This hypothesis is appropriate because it is testing whether there is a statistically significant difference in the mean time to complete all matches between the two matching schemes. The null hypothesis would be that there is no difference in the mean time between the two schemes, i.e., H₀:  - = 0.

To test this hypothesis, we can use a paired t-test, which compares the mean difference between the two sets of measurements (in this case, the time to complete all matches using the two matching schemes) to the standard error of the mean difference. If the t-test results in a p-value that is smaller than the chosen significance level (typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the mean times for the two matching schemes.

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The area of the gazebo is approximately 127.23 square feet.

To find the actual diameter of the gazebo, we need to use the scale of 2 inches = 5 feet. This means that for every 2 inches on the blueprint, the actual distance is 5 feet. Therefore, we can set up a proportion:

2 inches / 5 feet = 5.1 inches / x

where x is the actual diameter of the gazebo.

Solving for x, we get:

x = (5.1 inches * 5 feet) / 2 inches

x = 12.75 feet

So the actual diameter of the gazebo is 12.75 feet.

To find the area of the gazebo, we need to use the formula for the area of a circle:

A = π[tex]r^2[/tex]

where r is the radius of the circle. Since we know the diameter is 12.75 feet, the radius is half of that:

r = 12.75 feet / 2

r = 6.375 feet

Now we can plug in the radius to the formula for the area:

A = π(6.375 feet[tex])^2[/tex]

A = 127.23 square feet

So the area of the gazebo is approximately 127.23 square feet.

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In order to find the probability of the intersection of two events, P(A ∩ B), we can use the formula: P(A ∩ B) = P(A) + P(B) - P(A ∪ B) Therefore, the correct answer is c. 0.2100.

In order to find the probability of the intersection of two events, P(A ∩ B), we can use the formula:

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

Given the probabilities P(A) = 0.62, P(B) = 0.47, and P(A ∪ B) = 0.88, we can plug these values into the formula:

P(A ∩ B) = 0.62 + 0.47 - 0.88

P(A ∩ B) = 1.09 - 0.88

P(A ∩ B) = 0.21

Therefore, the correct answer is option (c) 0.2100.

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The answer is option(b) greater than equal to the approximation given by Chebyshev's theorem.

To answer this, we need to use the empirical rule and compare it with the approximation given by Chebyshev's theorem.

The empirical rule states that approximately 68% of the data will be within one standard deviation, 95% within two standard deviations, and 99.74% within three standard deviations of the mean, given that the data has a bell-shaped distribution.

In this case, we're looking at 3 standard deviations from the mean. According to the empirical rule, approximately 99.74% of the data will be within 3 standard deviations. Now, we compare the approximation given by the empirical rule (99.74%) to the approximation given by Chebyshev's theorem (89%).

Since 99.74% (empirical rule) is greater than 89% (Chebyshev's theorem), the approximation given by the empirical rule is greater than equal to the approximation given by Chebyshev's theorem.

Your answer: B. greater than equal to the approximation given by Chebyshev's theorem.

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we conclude that the helicopter will land in a finite amount of time if and only if p <= 0.

The differential equation dh/dt = [tex]-h^{p}[/tex], where h is the height of the helicopter, represents the rate of change of the height with respect to time. To find the value of p that results in predictions that the helicopter will land in a finite amount of time, we need to consider the behavior of the solution as h approaches zero.

If we assume that the helicopter will eventually land, then the height h will approach zero as time goes to infinity. Therefore, we can consider the behavior of the solution near the origin. To do this, we will use a technique called separation of variables.

Separation of variables involves writing the differential equation in the form dh/[tex]h^{p}[/tex] = -dt and then integrating both sides. This gives:

∫h_[tex]0^{h}[/tex] dh / [tex]h^{p}[/tex] = ∫0^t -dt

where h_0 is the initial height of the helicopter.

The left-hand side can be evaluated using the power rule of integration:

[tex][1/(1-p)] [h^{(1-p)}]_h_0^{h} = -t[/tex]

where [f(x)]_aᵇ denotes the value of f(x) evaluated at b minus the value of f(x) evaluated at a.

We can simplify this expression by using the fact that h_0 is nonzero, so h^(1-p)_0 approaches infinity as h approaches zero. Therefore, we can neglect the term h^(1-p)_0 and write:

[tex][1/(1-p)] h^{(1-p)} = -t[/tex]

If p > 1, then h^(1-p) approaches zero as h approaches zero. Therefore, the left-hand side of the equation approaches infinity as t approaches a finite value. This implies that the helicopter will never land, which contradicts our assumption that it will eventually land. Therefore, we must have p <= 1.

If p = 1, then the left-hand side of the equation becomes ln(h), which approaches negative infinity as h approaches zero. Therefore, the equation cannot hold for any finite value of t. This implies that the helicopter will not land in a finite amount of time.

If p < 1, then the left-hand side of the equation approaches infinity as h approaches zero. Therefore, the equation cannot hold for any finite value of t. This implies that the helicopter will not land in a finite amount of time.

Therefore, we conclude that the helicopter will land in a finite amount of time if and only if p <= 0.

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Answer:

Area = 2.1875

Step-by-step explanation:

The formula for area of a rhombus is A = 1/2(d1)(d2), where d1 is one of its rhombus and d2 is the other.  Thus, to find the area, we can plug into the formula 1.75 for d1 and 2.5 for d2 and solve for A:

A = 1/2(1.75)(2.5)

A = 0.875*2.5

A = 2.1875

The probability of randomly drawing a vowel is 2/5

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

W, E, L, O, V, E, M, A, T, H

Using the above as a guide, we have the following:

Vowels = 4

Total = 10

So, we have

P(Vowel) = Vowel/Total

Substitute the known values in the above equation, so, we have the following representation

P(Vowel) = 4/10 = 2/5

Hence, the solution is 2/5

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The longer leg of the triangle is 17✓3 yards based on the information of triangle and shorter leg length.

We will use law of sines relating an angle and it's opposite side -

sin shorter angle/shorter side = sin longer angle/longer side

The formula is as per the known fact that wide angle will have wider side opposite to it

Keep the values in formula -

sin 30/17 = sin 60/longer side

Longer side = 17 sin 60/sin 30

Substitute the values of sin 30 and sin 60

Longer side = (17 × ✓3/2)/(1/2)

Longer side = 17✓3 yards.

Thus, the longer leg of the right triangle is 17✓3 yards.

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The weight of the ice after 10 hours is 1217 lb

A 523 lb mass of ice melts at 4.3 hours

The first step is to calculate the weight after one hour

523= 4.3

x= 1

cross multiply both sides

4.3x= 523

x= 523/4.3

x= 121.7

The weight after 10 hours can be calculated as follows

121.7= 1

y= 10

y= 121.7 × 10

y= 1217

Hence the weight after 10 hours is 1217 lb

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Answer:

2(9(5) + 5(3) + 9(3)) = 2(45 + 15 + 27) = 2(87) = 174 square inches

Answer:[tex]\left(x-14\right)^{2}+\left(y-15\right)^{2}=92[/tex]

Step-by-step explanation:

Since the x value is 14 and the y value is 15, we know our H and K for this formula. In the formula, we state that the first part of the formula is equal to the radius squared. That would look something like this without the filled in values:

[tex]\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}[/tex]

The center point of the circle is found at (h, k)

I hope this helps :)

A quadrilateral is a shape with four sides: The indicated measures are A = 56, B = 128, C = 100 and D = 76.

The indicated measures:

The angles in a quadrilateral add up to 360 degrees.

So, we have:

4n + 5n + 6 + 9n + 2 + 8n - 12 = 360

Collect like terms

4n + 5n + 9n + 8n = 360 - 6 - 2 + 12

Evaluate the like terms

26n = 364

Divide through by 26

n = 14

From the figure, we have:

A = 4n

B = 9n + 2

C = 8n - 12

D = 5n + 6

So, we have:

A = 4 * 14 = 56

B = 9*14 + 2 = 128

C = 8*14 - 12 = 100

D = 5*14 + 6 = 76

Hence, the indicated measures are

A = 56, B = 128, C = 100 and D = 76

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Correct Question:

A window is the shape of a quadrilateral. Find the indicated measure

Answer:

[tex]\frac{62}{9}[/tex] or  6.8888889

Step-by-step explanation:

The explanation is on the attachment below

Answer:

65%

Step-by-step explanation:

ITS CORRECT

The Fourier series for the original function f(x) is:

f(x) = (1/2) + ∑[n=1,∞] [(4/nπ²) * (1 - (-1)^n) cos(nπx/2) + (4/nπ) * sin(nπ/2) sin(nπx/2)] for x € (-1,1)

To find the Fourier series coefficients for the given function, we need to first determine the period of the function.

Since the function is defined differently for x in the interval (-1,0) and (0,1), we can break down the function into two separate periodic functions, each with its own period.

For the interval (-1,0), the function is a constant function equal to 1. Hence, the period is simply 2.

For the interval (0,1), the function is a linear function given by f(x) = 2 - x. The period of a linear function is always infinite, but we can restrict the domain to a smaller interval to get a periodic function. We can choose the interval (0,2) as the period for this function, since f(x + 2) = 2 - (x + 2) = 2 - x = f(x) for all x in the interval (0,1).

Now, we can write the Fourier series for each of the two periodic functions:

For the function defined on (-1,0), the Fourier series coefficients are given by:

an = (1/2) * ∫[-1,1] f(x) cos(nπx/2) dx

= (1/2) * ∫[-1,0] cos(nπx/2) dx (since f(x) = 1 for x in (-1,0))

= (1/2) * [(2/π) * sin(nπ/2)]

bn = (1/2) * ∫[-1,1] f(x) sin(nπx/2) dx

= (1/2) * ∫[-1,0] sin(nπx/2) dx (since f(x) = 1 for x in (-1,0))

= (1/2) * [(2/π) * (1 - cos(nπ/2))]

For the function defined on (0,1), the Fourier series coefficients are given by:

an = (1/2) * ∫[0,2] f(x) cos(nπx/2) dx

= (1/2) * ∫[0,1] (2 - x) cos(nπx/2) dx

= (1/2) * [(4/nπ²) * (1 - (-1)^n)]

bn = (1/2) * ∫[0,2] f(x) sin(nπx/2) dx

= (1/2) * ∫[0,1] (2 - x) sin(nπx/2) dx

= (1/2) * [(4/nπ) * sin(nπ/2)]

Hence, the Fourier series for the original function f(x) is:

f(x) = (1/2) + ∑[n=1,∞] [(4/nπ²) * (1 - (-1)^n) cos(nπx/2) + (4/nπ) * sin(nπ/2) sin(nπx/2)] for x € (-1,1)

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Answer:

16 science problems

26 math problems

Step-by-step explanation:

m = number of math problems

s = number of science problems

m = s + 10

m + s = 42

(s + 10) + s = 42

2s + 10 = 42

2s = 42 - 10 = 32

s = 32/2 = 16

m = s + 10 = 16 + 10 = 26

The hexagon's apothem is approximately 13.856 inches long. The hexagon's perimeter is 96 inches. The hexagon has a surface area of approximately 665.088 square inches.

A hexagon is a six-sided polygon in geometry. The sum of any simple (non-self-intersecting) hexagon's internal angles is 720°.

Given that the length of a side is = 16 in

So half a side = 8 in

Using the Pythagorean theorem, calculate the area of the given right triangle.

Apothem = [tex]\sqrt{16^{2} - 8^{2} }[/tex]

= [tex]\sqrt{256 - 64}[/tex]

= √192

= 13.856 inches.

Now, we will calculate the perimeter of the hexagon. We have been given 6 sides of hexagon and each side length is 16 in, so

Perimeter = 16 × 6 = 96 inches

Area of hexagon = 1/2 × apothem × perimeter

= 1/2 × 13.856 × 96

= 665.088 inches

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Correct question:

Find the missing measures in this regular hexagon.

A regular hexagon has side lengths of 16 inches. The radius is 16 inches. An apothem is shown.

The length of the apothem of the hexagon is about ___inches.

The perimeter of the hexagon is ___ inches.

The area of the hexagon is about ___ square inches.

In a direct proof, evidence is used to support a claim, On the other hand, a counterexamples is a single example that show the contradictions in a claim.

Evidence means any piece of information that supports the argument being made in a proof which could include mathematical formulas, logic, or theorems that have been previously proven.

Counterexamples are specific examples that disprove a statement made in a proof and are used to show that a proof is not valid and that the argument being made is flawed.

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The value of the variable x is 24

To determine the value of the variable, it is important that we know;

From the information given, we have;

<NLM ≅ <NOP

We have the values;

NLM = x + 12

NOP = 20 + 16

Now, substitute the values

x + 12 = 20 + 16

add the values

x + 12 = 36

collect the like terms

x = 36 - 12

subtract the values

x = 24

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The standard error of the mean is 1.16. At 95% confidence, the margin of error is 2.27.

(a) The standard error of the mean, σx, can be calculated using the formula:
σx = σ/√n
where σ is the population standard deviation, and n is the sample size.
Substituting the given values, we get:
σx = [tex]\frac{9}{ \sqrt{60} }[/tex]
σx = 1.16
Therefore, the standard error of the mean is 1.16.

(b) To find the margin of error, we need to use the formula:
The margin of error = z(σx)
where z is the z-score corresponding to the level of confidence. For 95% confidence, the z-score is 1.96 (using a standard normal distribution table).
Substituting the values we get:
Margin of error = 1.96(1.16)
Margin of error = 2.27

Therefore, at 95% confidence, the margin of error is 2.27. This means that we can be 95% confident that the true population means falls within 2.27 units of the sample mean of 25.

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The dimension of the rectangular prism is ( 2x - 1) which has a volume 4x⁴ + 14x³ - 8x²

The given volume of the rectangular prism is,

V = 4x⁴ + 14x³ - 8x²

Let the length, breadth and height of the rectangular prism be l, b, and h respectively.

Thus, by formula of volume of a rectangular prism we get,

l*b*h = 4x⁴ + 14x³ - 8x²

⇒ l*b*h = 2x² ( 2x² + 7x - 4)

= 2x² [ 2x² + 8x - x - 4]

= 2x² [ 2x( x + 4) -1( x + 4) ]

= 2x² ( 2x - 1 )( x + 4 )

Therefore, by equating the above equation, obtained from simplifying the equation of volume of a rectangular prism, with zero , we get,

2x² ( 2x - 1 )( x + 4 ) = 0

⇒ 2x² = 0 ⇒ x = 0

and, ⇒ 2x - 1 = 0 ⇒ x = 1/2

and, ⇒ x + 4 = 0 ⇒ x = -4

Thus we can see that only equation (2x - 1) gives a possible value of x, that is either the length or breadth or height of the rectangular prism.

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