An administrator of a large middle school is installing some vending machines in the school. She wants to know what type of machine would be most popular.

the series converges for x in the interval: (-1/6, 1/6)

The series ∑(6)^nx^n converges if |x| < 1/6. This can be determined using the ratio test, where we take the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term:

|6(x^(n+1))/(x^n)| = 6|x|

As n approaches infinity, this limit is less than 1 if and only if |x| < 1/6. Therefore, the series converges for all x in the open interval (-1/6, 1/6).

In interval notation, we can write the answer as: (-1/6, 1/6).
The given series is:

∑(6^n)(x^n)

This is a geometric series with a common ratio of 6x. For a geometric series to converge, the absolute value of the common ratio must be less than 1:

|6x| < 1

To find the values of x for which the series converges, we can solve for x in the inequality:

-1 < 6x < 1

Divide by 6:

-1/6 < x < 1/6

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The graph that represents a function is the graph (b)

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

Graphs A to D

As a general rule of the vertical line test

For a graph to represent a function, a line drawn from the x-axis must intersect with the graph at most once

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

The graph b would intersect with a line from the x-axis at most once

Hence, the graph that represents a function is (b)

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The company’s earnings per share reported in its financial statements for the year ended December 31, 2020 is $1.45. The correct option is c.

To calculate earnings per share, we need to take the company's net income and divide it by the weighted average number of shares outstanding during the year.

First, let's adjust for the stock dividend on July 31. Since the dividend was 100%, we can double the number of shares outstanding to 416,000:

Jan. 1: 200,000 shares
Feb. 1: 8,000 new shares
July 31: 200,000 shares doubled to 400,000 shares
Dec. 31: 416,000 shares

Next, we need to calculate the weighted average number of shares outstanding during the year. We can do this by taking the number of shares outstanding for each period and multiplying it by the number of months those shares were outstanding:

Jan. 1 to Jan. 31: 200,000 shares x 1 month = 200,000
Feb. 1 to July 31: (200,000 + 8,000) shares x 6 months = 1,248,000
Aug. 1 to Dec. 31: 416,000 shares x 5 months = 2,080,000

Total weighted average shares outstanding: 3,528,000

Finally, we can divide the net income of $560,000 by the weighted average shares outstanding of 3,528,000 to get earnings per share of $0.1585. Multiplying this by 9 (since there are 9 months of the year remaining after February 1) gives us earnings per share of $1.4265. Rounded to the nearest penny, the answer is $1.45.

Thus, The correct option is c.

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The fourth equation i.e. cos (60°) = [tex]\frac{a}{12}[/tex] can be used to find one of the missing side lengths in the triangle. This has been obtained using trigonometry.

What is trigonometry?

Trigonometry is a branch of mathematics that focuses on right-angled triangles, including their sides, angles, and connections.

We are given a right angled triangle with perpendicular as b, base as a and hypotenuse measuring 12 units.

We know by trigonometry that cos θ is the proportion of the adjacent side to the hypotenuse.

So, only the fourth equation is the one which can be used to find the missing length.

So, the equation is cos(60°) = [tex]\frac{a}{12}[/tex].

Hence, the fourth option is the correct answer.

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a. True, more than 5% of the children selected "other".

b. True, 15% of kids say that Lucky Charms is their favourite cereal.

c. As the cereal that the most students chose, Frosted Flakes, it is true that this is the most popular cereal.

d. Fruit Loops were picked half as frequently as Frosted Flakes.

Using Percentage calcuate number of children:

Total number of children = 15 + 10 + 2 + 20 + 3 = 50

a). 3 children chose other.

Percentage = 3/50 × 100

= 6%

So it's accurate to say that more than 5% of the kids selected "other".

b). 15 children chose "lucky charm".

Percentage = 15/50 × 100

= 30%.

So it is true that 15% of kids prefer Lucky Charms cereal.

c). Frosted Flakes are the cereal that the majority of pupils choose, so it is true that this is the case.

d). Ten kids all chose Fruit Loops.

Twenty kids chose Frosted Flakes.

As a result, Frosted Flakes were picked twice as often as Fruit Loops.

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It would take Annmarie and Gladys 96 minutes to plow the field together.

Annmarie can plow a field in 240 minutes. Gladys can plow the same field 80 minutes faster than Annmarie.

So Gladys can plow the field in 240 - 80 = 160 minutes.

Let x be the time it takes for both of them to plow the field together.

The combined rate of Annmarie and Gladys is the sum of their individual rates.

Annmarie's rate is 1 field per 240 minutes, which is 1/240 field per minute.

Gladys's rate is 1 field per 160 minutes, which is 1/160 field per minute.

Their combined rate is:

1/240 + 1/160 = 1/x

Simplifying this equation:

1/x = (4/960) + (6/960) = 10/960

1/x = 1/96

Multiplying both sides by 96, we get:

x = 96 minutes

Therefore, it would take Annmarie and Gladys 96 minutes to plow the field together.

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We know that the length of TU to the nearest tenth is 6.9 in

The figure is tangent to the circle at point U. Using the information given in the figure, we can conclude that segment ST is tangent to the circle at point T.

To find the length of TU, we can use the formula for the length of a tangent segment from a point outside the circle:

TU^2 = TS x TR

We know that TS = ST = 4 in. To find TR, we can use the Pythagorean theorem:

TR^2 = RS^2 - TS^2
TR^2 = 8^2 - 4^2
TR^2 = 48
TR = sqrt(48)

Now we can substitute the values we have found into the first formula:

TU^2 = 4 x sqrt(48)
TU = sqrt(4 x sqrt(48))
TU ≈ 6.9 in.

Therefore, the length of TU to the nearest tenth is 6.9 in.

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Answer:

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 9}

Step-by-step explanation:

We can find the union of two sets by including all of the numbers in both sets, but without repeating any numbers.

For example:

if A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8},

then A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

We can apply this concept to the problem at hand, but first we need to represent set A as a list of numbers:

They all have to be odd numbers between 0 and 10.

    [tex]\implies A[/tex] = {1, 3, 5, 7, 9}

We are given that B = {2, 3, 4, 5, 6}. So, to find the union of A and B, we can combine both sets of numbers and get rid of copies:

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 9}

There are 86,400 ways to arrange 5 different green balls and 7 different red balls in a row if all the green balls are separated.

If all the green balls are separated, we can think of the green balls as dividers that separate the red balls into groups. Since there are 5 green balls, there will be 6 groups of red balls. For example, if there are 7 red balls, the arrangement might look like this:

| R R R R R R R |

The "|" symbols represent the green balls. Each group of red balls is between two green balls.

To count the number of arrangements, we can think of each group of red balls as a box, and the green balls as dividers between the boxes.

We can arrange the 6 boxes in a row in 6! = 720 ways, and we can arrange the 5 green balls in the remaining 5 positions in 5! = 120 ways. Therefore, the number of arrangements is:

6! x 5! = 720 x 120 = 86,400

So ,there are 86,400 ways to arrange 5 different green balls and 7 different red balls in a row if all the green balls are separated.

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Kareem drove the truck for 150 miles.

As areem rented a truck for one day. there was a base fee of $19.99 , and there was an additional charge of 83 cents for each mile driven we will let the number of miles driven by Kareem be represented by 'x'. The total cost, including the base fee, can be represented as:

Total cost = Base fee + Additional charge per mile * Number of miles driven

$135.36 = $19.99 + $0.83x

Subtracting $19.99 from both sides and then dividing by $0.83, we get:

x = (135.36 - 19.99) / 0.83

x = 150

Therefore, Kareem drove the truck for 150 miles.

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a.  The ratio of Es to Ns in percentage is 64%

b. Probability that it will take at least five patients to find one patient on whom the medicine would not be effective is impossible to estimate this probability from the table.

c. The probability that the medicine will be effective on exactly three out of four randomly selected patients is  12.9%

a. The total number of patients in the table is 50.

To calculate the ratio of Es to Ns in percentage, we count the number of Es and Ns and divide the number of Es by the number of Ns and Es combined, and then multiply by 100. Counting the table, we find that there are 32 Es and 18 Ns. So, the ratio of Es to Ns in percentage is:

32 / (32 + 18) * 100 = 64%

b. To estimate the probability that it will take at least five patients to find one patient on whom the medicine would not be effective, we need to look at the runs of Ns in the table. We can see that there are no runs of five or more Ns, so it is impossible to estimate this probability from the table.

c. To estimate the probability that the medicine will be effective on exactly three out of four randomly selected patients, we need to count the number of ways we can choose three Es and one N, and divide by the total number of possible outcomes of selecting four patients from the table. The total number of possible outcomes is:

50 choose 4 = 50! / (4! * (50-4)!) = 230300

The number of ways we can choose three Es and one N is:

32 choose 3 * 18 choose 1 = (32! / (3! * (32-3)!)) * (18! / (1! * (18-1)!)) = 32 * 31 * 30 / (3 * 2) * 18 = 32 * 31 * 30 * 18 / 6 = 297120

So, the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients is:

297120 / 230300 ≈ 0.129 or about 12.9% (rounded to the nearest tenth of a percent).

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We can make 24 bouquets of flowers for a wedding, each with 10 roses and 7 lilies.

To determine the largest number of identical bouquets that can be made using 240 roses and 168 lilies, we need to find the greatest common factor (GCF) of these two numbers.

The prime factorization of 240 is 2^4 x 3 x 5, while the prime factorization of 168 is [tex]2^3 * 3 * 7[/tex]. To find the GCF, we can take the product of the common prime factors raised to the smallest exponent they appear in either number. Therefore, the GCF of 240 and 168 is [tex]2^3 * 3[/tex] = 24.

This means that we can make 24 identical bouquets using 240 roses and 168 lilies. To do so, we would use 10 roses and 7 lilies in each bouquet, since 10 and 7 are the largest numbers that divide both 240 and 168 without remainder, respectively. So, we can make 24 bouquets, each with 10 roses and 7 lilies.

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The percent of sales that occurred in May is: 6.6%

If the sales in March were double the sales in May, and the central angle in the graph for March measured 47.5°, we can find the central angle for May as follows:

Let x be the central angle for May. Then we have:

2x = 47.5

Solving for x, we get:

x = 23.75

So the central angle for May is 23.75°.

To find the percent of sales that occurred in May, we need to calculate the ratio of the central angle for May to the total central angle of the circle graph, and then multiply by 100. The total central angle of a circle is always 360°.

So the percent of sales that occurred in May is:

(23.75/360) x 100 = 6.6%

Therefore, 6.6% of the sales occurred in May.

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The exact value for each trigonometric function are -12/13, -5/13 and 12/5

To find the reference angle, we can use the properties of right triangles. We can draw a line from the point (-5, 12) to the x-axis to form a right triangle. The hypotenuse of the triangle is the distance from the point (-5, 12) to the origin, which is the square root of the sum of the squares of the x and y coordinates:

√((-5)² + 12²) = 13

The reference angle is the acute angle between the x-axis and the adjacent side of the triangle, which is the x-coordinate of the point (-5, 12) divided by the hypotenuse:

cosθ = -5/13

θ = arccos(-5/13)

θ ≈ 2.214 radians

The angle's standard form is given by the equation:

θ = n(2π) ± α

where n is an integer, and α is the angle's reference angle. Since the point (-5, 12) is in the second quadrant, the angle's terminal side intersects the unit circle at an angle of θ = π + α. Therefore, the standard form of the angle is:

θ = (2n + 1)π - arccos(-5/13)

To find the exact value of the trigonometric functions of this angle, we can use the properties of the unit circle. Since the sine function is positive in the second quadrant, we have:

sinθ = sin(π + α) = -sinα = -12/13

Similarly, since the cosine function is negative in the second quadrant, we have:

cosθ = cos(π + α) = -cosα = -5/13

Finally, since the tangent function is the ratio of the sine and cosine functions, we have:

tanθ = tan(π + α) = -tanα = 12/5

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The volume of the cone is approximately one-third the volume of the

cylinder

:V(cone) ≈ 1/3 V(cylinder) = 1/3 (9.4) ≈ 3.1 cubic feet.

The volume of a cone with the same height and radius as the given

cylinder, we can use the formula:

[tex]V = 1/3 πr²h[/tex]

where V is the volume, r is the radius, and h is the height.

Since the cylinder has a volume of approximately 9.4 cubic feet, we can

use the formula for the volume of a cylinder

:V = πr²hand solve for the radius:

[tex]r = √(V/πh) = √(9.4/πh)[/tex]

Now we can substitute this expression for the radius into the formula for

the volume of a cone:

[tex]V = 1/3 π(√(V/πh))²h[/tex]

Simplifying this expression gives:

[tex]V = 1/3 π(V/πh)hV = 1/3 V[/tex]

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The correct answer to this question is C, 100ft/min.

The rate of climbing can be calculated by dividing the increase in elevation (100 ft) by the time it takes to make that increase (2 min).

This gives a rate of 50ft/min. Therefore, the class is climbing at a rate of 100ft/min since the question asks for the rate of their climbing.

It's important to pay attention to the wording of the question to ensure that the correct answer is selected. In this case, the question specifically asks for the rate of climbing, not the rate of increase in elevation.

Always read the question carefully and make sure to include all given information when solving problems with content loaded.

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Answer:

  a) 149,270 m³

  b) 4 hours 9 minutes

Step-by-step explanation:

You want to know the volume of air in a 4 km tunnel 8 m high and 5 m wide with a semicircular cross section at the top. And you want to know the replacement time for that air if it is exchanged at 10 m³ per second.

The area of the semicircular top of the tunnel is ...

  A = π/8d²

  A = π/8·(5 m)² = 25π/8 m²

The area of the rectangular bottom of the tunnel is ...

  A = LW

  A = (5 m)(8 -2.5 m) = 27.5 m²

So the total cross sectional area of the tunnels is ...

  (25π/8 +27.5) m² ≈ 37.317477 m²

The volume of the tunnel is ...

  V = Bh

  V = (37.317477 m²)(4000 m) ≈ 149269.9 m³

The volume of the air in the tunnel is about 149269.9 m³.

At 10 m³ per second, the air will be replaced after ...

  (149269.9 m³)/(10 m³/s) = 14926.99 s ≈ 4 hours 9 minutes

The given pump takes 4 hours 9 minutes to replace the air in the entire tunnel.

__

Additional comment

This tunnel would not meet any standard of safety.

A typical building is supposed to have about 5 air exchanges per hour. That's about 21 times the rate given here. Some tunnel structures need to have the air exchanged once per minute. (Here, that would mean the wind in the tunnel is 149 mph.)

The breathing requirements of the tunnel occupants need to be considered, as well as removal of air pollutants.

There are 3600 seconds in 1 hour.

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The rent of the apartment during the 10th year of living in the apartment would be $4940.79

To solve the problem, we need to use the formula for compound interest:

A = P * (1 + r/n)^(n*t)

where:

A is the final amount

P is the initial principal

r is the annual interest rate (as a decimal)

n is the number of times the interest is compounded per year

t is the number of years

In this case, the "principal" is the initial rent of $1600 per month, and the "interest rate" is the 12.5% increase per year. We want to find out the rent during the 10th year, so t = 10.

Let's plug in the values and solve for A:

A = 1600 * (1 + 0.125/12)^(12*10)

A = 1600 * (1.01042)^120

A = 1600 * 3.086

A = 4940.79

So the rent during the 10th year would be $4940.79 per month, rounded to the nearest tenth.

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Answer:

Method 1 (Using Slope-Intercept Form):

First, we need to find the equation of the line that passes through the given points.

Slope (m) = (Change in y) / (Change in x) = (45 - 25) / (4 - 0) = 20 / 4 = 5

Using the slope and one point (0, 25), we can find the y-intercept:

y - y1 = m(x - x1)

y - 25 = 5(x - 0)

y = 5x + 25

Therefore, when the input is 200, the output would be:

y = 5(200) + 25

y = 1025

Method 2 (Using Linear Interpolation):

We can use the formula for linear interpolation:

y = y1 + ((x - x1) / (x2 - x1)) * (y2 - y1)

where:

x1 = 0, y1 = 25

x2 = 4, y2 = 45

x = 200

Substituting the values, we get:

y = 25 + ((200 - 0) / (4 - 0)) * (45 - 25)

y = 25 + (200 / 4) * 20

y = 25 + 500

y = 525

Therefore, when the input is 200, the output would be approximately 525.

Answer:

(x-6)^2+(y-4)^2=10

Step-by-step explanation:

use the equation of a circle ( x- x coordinate of center)^2+(y-y coordinate of the center)^2=radius squared.

to solve for the radius, make a right triangle from the center to the point on the circle. 1^2+3^2=r^2. So r^2 is 10

Answer: (18, 4)

Part 1:

The cost of seeds and dried fruit together is $22:

1.5x + 2.5y = 22

Hunter plans to spend all of his money on seeds and dried fruit:

x + y = 22

Part 2:

We can use substitution method to solve the system of equations:

x = 22 - y   (from the second equation)

1.5(22 - y) + 2.5y = 22   (substitute x into the first equation)

33 - 1.5y + 2.5y = 22

y = 4

Substituting y = 4 into x + y = 22, we get x = 18.

Therefore, the ordered pair representing the number of seeds and dried fruit that Hunter bought is (18, 4).

Based on study, it appears that mind-set does matter for maids in terms of weight loss - simply thinking they are exercising more may have led to greater weight loss in the informed group.

To determine whether mind-set matters in terms of weight loss for the maids, we need to conduct a hypothesis test.

Null Hypothesis: The average weight loss for the informed group of maids is equal to the average weight loss for the uninformed group of maids.

Alternative Hypothesis: The average weight loss for the informed group of maids is greater than the average weight loss for the uninformed group of maids.

We can use a one-sided t-test to test this hypothesis, since we are interested in whether the informed group lost more weight than the uninformed group.

Using the data provided, we can calculate the sample mean and standard deviation for each group:

Informed group:

Sample size (n) = 44

Sample mean = 2.00 lbs

Sample standard deviation = 2.50 lbs

Uninformed group:

Sample size (n) = 76

Sample mean = 1.33 lbs

Sample standard deviation = 2.31 lbs

We can use a t-test with unequal variances (since the sample standard deviations are different) to test the hypothesis. Using a significance level of 0.05 and a one-tailed test, the critical t-value is 1.67 (from a t-distribution with 118 degrees of freedom).

The calculated t-value is: t = (2.00 - 1.33) / sqrt((2.50^2/44) + (2.31^2/76)) = 1.80

Since the calculated t-value (1.80) is greater than the critical t-value (1.67), we reject the null hypothesis and conclude that there is evidence that the informed group of maids lost more weight than the uninformed group of maids.

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The TWO statements that represent the relationship between y = 5x and y = log5 x are:

1. They are the exponential and logarithmic form of the same equation.

2. They are inverses of one another.

The two equations are related because they represent the same relationship between x and y, but in different forms. The first equation is an exponential equation, where y is a power of 5 raised to the x power. The second equation is a logarithmic equation, where y is the exponent to which 5 must be raised to get x.

Because the two equations represent the same relationship, they are inverses of one another. If we take the logarithm of both sides of the exponential equation, we get the logarithmic equation. If we raise 5 to both sides of the logarithmic equation, we get the exponential equation. Therefore, the two equations are inverses of one another.

70 percent of the guests at this hotel used the valet service.

To find the percentage of guests who used the valet service, we can follow these steps:

1. Add the number of guests who used the valet service (280) and those who used the public parking garage (120) to find the total number of guests with cars: 280 + 120 = 400 guests.

2. Divide the number of guests who used the valet service (280) by the total number of guests with cars (400).

3. Multiply the result by 100 to convert it into a percentage.

So, let's calculate the percentage:

(280 / 400) * 100 = 0.7 * 100 = 70%

Thus, 70% of the guests at this hotel used the valet service.

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The radius of an eyebrow window with a width of 62.8 inches and a height of 18.5 inches is approximately 36.7 inches.

To find the radius of an eyebrow window, we first need to understand its shape. An eyebrow window is a type of arched window that has a curved shape similar to that of an eyebrow. The shape of an eyebrow window is created by a combination of a circular arc and a straight line.

To find the radius of an eyebrow window with a width of 62.8 inches and a height of 18.5 inches, we need to use some geometry formulas. The height of the eyebrow window represents the height of the circular arc, and the width represents the diameter of the circle.

The formula for the radius of a circle is r = d/2, where r is the radius and d is the diameter. To find the diameter, we divide the width by pi (3.14). So, the diameter is 62.8/3.14 = 20 inches.

The height of the circular arc is half of the width, which is 18.5/2 = 9.25 inches. To find the radius, we use the formula for the height of a circular arc, h = r(1-cos(a/2)), where h is the height, r is the radius, and a is the angle of the arc.

The angle of the arc can be found using trigonometry. The sine of half the angle is equal to the height divided by the radius. So, sin(a/2) = h/r. Solving for a, we get a = 2arcsin(h/r).

Plugging in the values, we get a = 2arcsin(9.25/r). To find the radius, we solve for r using a calculator or algebra. The radius is approximately 36.7 inches.

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In a bag containing 5 red, 3 blue, and 12 yellow marbles, we will predict the number of times Hazel will select a blue marble out of 500 trials.

Step 1: Calculate the total number of marbles in the bag:
Total marbles = 5 red + 3 blue + 12 yellow = 20 marbles

Step 2: Determine the probability of selecting a blue marble:
Probability of selecting a blue marble = (number of blue marbles) / (total marbles) = 3 blue / 20 marbles = 3/20

Step 3: Predict the number of times Hazel will select a blue marble in 500 trials:
Predicted blue marbles selected = (probability of selecting a blue marble) x (total trials) = (3/20) x 500

Step 4: Perform the calculation:
(3/20) x 500 = 75

In conclusion, we predict that Hazel will select a blue marble 75 times out of 500 trials, given that the bag contains 5 red, 3 blue, and 12 yellow marbles.

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Answer:

24π cm²

Concepts Applied:

SA (TSA) of a cone = π · r · ( l+r )

Relation between l, h, and r i.e. l²=h²+r²

(h: cone height, r: base radius, l: slant height)

Step-by-step explanation:

Calculating the Slant height:

l²=h²+r²

l = sqrt(h²+r²)

l = sqrt(16+9)

l = sqrt(25)

l = +5 cm (distance is a scalar quantity)

Calculating the TSA:

= π · 3 · (5+3)

= 24π cm²

Answer:

34π cm^2 is the correct answer

We can conclude that Arnold's height of 4.75 feet by using tangent function is not accurate based on the calculations.

To determine whether Arnold's height is accurate, we can use trigonometry and the given information about the angle of elevation and distance to the building.

Let's start by drawing a diagram to represent the situation. We have a right triangle with the opposite side being Arnold's height (h), the adjacent side being the distance to the building (300 ft), and the angle of elevation being 22 degrees.

Using the tangent function, we can write:

tan(22) = h / 300

Solving for h, we get:

h = 300 * tan(22)

Using a calculator, we find that h is approximately 122.8 feet.

Therefore, we can conclude that Arnold's height of 4.75 feet is not accurate based on the calculations. The height we calculated is over 25 times greater than Arnold's claimed height, which is not possible.

It is important to note that this assumes that Arnold's line of sight is parallel to the ground. If Arnold is on uneven ground, this could affect the calculations. Additionally, it is possible that Arnold may have misspoken about his height or the angle of elevation. However, based on the given information and calculations, we can confidently say that Arnold's claimed height is not accurate.

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Based on experimental probability, Abdul should expect 10 tails in the next 40 flips.

The experimental probability of getting tails is calculated as the ratio of the number of tails obtained in the experiment to the total number of coin flips.

In this case, the experimental probability of getting tails is 16/64, which simplifies to 1/4 or 0.25.

This means that in the long run, we would expect approximately 25% of coin flips to result in tails.

To determine how many of the next 40 flips Abdul should expect to come up tails based on experimental probability,

we can multiply the experimental probability by the total number of flips.

Abdul should expect 0.25 × 40 = 10 tails in the next 40 flips.

Experimental probability of getting tails = Number of tails obtained / Total number of coin flips

Experimental probability of getting tails = 16/64

Experimental probability of getting tails = 0.25 or 25%

Expected number of tails in the next 40 flips = Experimental probability of getting tails × Total number of flips

Expected number of tails in the next 40 flips = 0.25×40 = 10 tails

Learn more about probability here:

https://brainly.com/question/11234923

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