Full Boat Manufacturing has projected sales of $115. 5 million next year. Costs are expected to be $67. 4 million and net investment is expected to be $12. 3 million. Each of these values is expected to grow at 9 percent the following year, with the growth rate declining by 1 percent per year until the growth rate reaches 5 percent, where it is expected to remain indefinitely. There are 4. 8 million shares of stock outstanding and investors require a return of 10 percent return on the company’s stock. The corporate tax rate is 21 percent

The correct statement regarding the solution to the system of equations is given as follows:

b. Infinite solutions.

The system of equations in the context of this problem is defined as follows:

Replacing the second equation into the first, the value of x is obtained as follows:

3(8 + 2x) - 6x = 24

24 + 6x - 6x = 24

24 = 24.

24 = 24 is a statement that is always true, hence the system has an infinite number of solutions, and thus option B is the correct option for this problem.

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The result of subtracting 5 from 3 1/3 is -2 2/3.

To subtract 5 from 3 1/3, we need to first convert the mixed number to an improper fraction. This can be done by multiplying the whole number (3) by the denominator of the fraction (3), and adding the numerator (1) to get 10/3. Therefore, 3 1/3 is equivalent to 10/3.

Next, we can subtract 5 from 10/3 by finding a common denominator of 3, which gives 15/3 - 10/3 = 5/3. This is the result in improper fraction form.

To convert back to a mixed number, we can divide the numerator (5) by the denominator (3), which gives a quotient of 1 and a remainder of 2. Therefore, the answer is -2 2/3.

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(a) The function rule for Big Wave Waterpark, f(x) = 2.5x + 5. where f(x) is total cost and x is number of slides ridden.

b) The function rule for Coaster City is, f(x) = 5x + 7.50, where x is the number of roller coaster ridden and f(x) is total cost.

c) The function rule for Virtual Reality Lan is, f(x) = 3x + 10, where f(x) is total cost and x is the number reality rides ridden.

(a) Let f(x) = ax +b be the function which represents the total cost of Big Wave Park where x represents the number of taken ride.

We can see that f(2) = 10; f(4) = 15 and f(6) = 20.

Therefore, 2a + b = 10 and 4a + b = 15

So, 2(2a + b) - (4a + b) = 2*10 - 15

4a + 2b - 4a - b = 20 - 15

b = 5

Now, f(6) = 20

6a + b = 20

6a + 5 = 20 [putting the value of 'b']

6a = 20 - 5 = 15

a = 15/6 = 5/2 = 2.5

Hence, the function rule for Big Wave Waterpark is, f(x) = 2.5x + 5.

(b) The function rule for Coaster city is, f(x) = 5x + 7.50, where x is the number of roller coaster ridden.

(c) Let the total cost for Virtual Reality Lan is, f(x) = cx + d, where x is the number reality rides ridden.

From the given graph we can see that, f(10) = 40; f(20) = 70; f(30) = 100.

So, 10c + d= 40 ........... (i)

20c + d = 70 ............... (ii)

Solving (i) and (ii) we get,

2(10c + d) - (20c + d) = 2*40 - 70

20c + 2d - 20c - d = 80 - 70

d = 10

So putting the value d = 10 in f(30) = 100 we get,

30c + 10 = 100

30c = 100 - 10 = 90

c = 90/30 = 3

So the function rule is, f(x) = 3x + 10.

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The correct answer is D. 29.50.

To find the wingspan of the actual airplane, we will use the given scale factor and the wingspan of the model airplane.
Given information:
Model airplane wingspan = 11.8 inches
Scale factor = 1 inch to 2.5 feet

Step 1: Convert the model wingspan to feet using the scale factor.
1 inch on the model represents 2.5 feet on the actual airplane. To convert 11.8 inches to feet, multiply by the scale factor.
Step 2: Calculate the actual wingspan.
Actual wingspan = Model wingspan * Scale factor
Actual wingspan = 11.8 inches * 2.5 feet/inch

Step 3: Perform the multiplication.
Actual wingspan = 29.5 feet
So, the wingspan of the actual airplane is 29.5 feet.

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Answer:

Since Q is a center of gravity, we can apply these formulas

The probability of a hurricane hitting the U.S. coast and being a category 1 hurricane is 0.403, or 40.3%.

The National Oceanic and Atmospheric Administration (NOAA) is a federal agency that is responsible for monitoring and predicting changes in the Earth's environment, including the atmosphere and oceans. One of their main responsibilities is to track and study hurricanes that affect the United States.

According to their data, there have been a total of 290 hurricanes that have hit the U.S. coast between 1851 and 2013. These hurricanes are categorized based on their wind speeds, with categories ranging from 1 to 5.

To create a probability distribution for this data, we need to calculate the probability of each category of hurricane occurring. We can do this by dividing the number of hurricanes in each category by the total number of hurricanes.

Category 1 hurricanes: 117/290 = 0.403
Category 2 hurricanes: 76/290 = 0.262
Category 3 hurricanes: 76/290 = 0.262
Category 4 hurricanes: 18/290 = 0.062
Category 5 hurricanes: 3/290 = 0.010

Therefore, the probability of a hurricane hitting the U.S. coast and being a category 1 hurricane is 0.403, or 40.3%. This means that out of all the hurricanes that have hit the U.S. coast, about 40% of them were category 1 hurricanes. It is important to note that this probability distribution is based on historical data and may not accurately predict the likelihood of future hurricanes.

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Brandon rode his bike 8.35 miles further than Randy.

To find out how much further Brandon rode his bike last month than Randy, you'll need to subtract Randy's miles from Brandon's miles using the given values.

Step 1: Identify the miles ridden by both Brandon and Randy.
- Brandon rode 56.28 miles.
- Randy rode 47.93 miles.

Step 2: Subtract Randy's miles from Brandon's miles.
- 56.28 miles (Brandon's miles) - 47.93 miles (Randy's miles) = 8.35 miles.

So, last month, Brandon rode his bike more than Randy.

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In triangle EOD, angle EOD = 14 degrees, angle DOE = 76 degrees, and angle DEO = 90 degrees. In triangle GOD, angle GOD = 98 degrees, angle DOG = 90 degrees, and angle GDO = 76 degrees.

Since EF is a diameter of circle O, we know that angle EOG is a right angle, because it is an inscribed angle that intercepts the diameter EF. Therefore, angle EOG = 90 degrees.

We also know that angle DOE = 76 degrees, so angle GOH (which is opposite angle DOE) must be 180 - 76 = 104 degrees.

Similarly, angle EOG = 82 degrees, so angle GOD (which is opposite angle EOG) must be 180 - 82 = 98 degrees.

Now, we can use the fact that angles in a triangle add up to 180 degrees to find angle DOG:

angle DOG = 180 - angle GOD - angle GOH

= 180 - 98 - 104

= -22

This result doesn't make sense, because angles can't be negative. However, we made a mistake when calculating angle GOH earlier. Since D and G are opposite sides of EF, they must be collinear.

Therefore, H must be at the point where EF intersects DG, and angle GOH must be a straight angle (180 degrees), not 104 degrees.

With this correction, we have:

angle GOH = 180 degrees

angle GOD = 98 degrees

angle DOG = 180 - angle GOD - angle GOH

= 180 - 98 - 180

= -98

Again, this result doesn't make sense because angles can't be negative. We made another mistake when calculating angle DOG.

Since EF is a diameter of circle O, angles DOG and DEG must be right angles. Therefore, we have:

angle DOG = 90 degrees

angle DEG = 90 degrees

Finally, we can use the fact that angles on a straight line add up to 180 degrees to find angle EOD:

angle EOD = 180 - angle DOG - angle DOE

= 180 - 90 - 76

= 14

Therefore, the angles in triangle EOD are:

angle EOD = 14 degrees

angle DOE = 76 degrees

angle DEO = 90 degrees

And the angles in triangle GOD are:

angle GOD = 98 degrees

angle DOG = 90 degrees

angle GDO = 180 - angle GOD - angle DOG

= 180 - 98 - 90

= -8

Once again, we have a negative angle, which doesn't make sense.

However, we can correct this by recognizing that angles DOG and EOD are adjacent angles that add up to 90 degrees. Therefore, we have:

angle GDO = 90 degrees - angle EOD

= 90 - 14

= 76 degrees

Therefore, the angles in triangle GOD are:

angle GOD = 98 degrees

angle DOG = 90 degrees

angle GDO = 76 degrees

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Our calculated t-value (-2.54) is beyond this critical value, we can reject the null hypothesis and conclude that there is a significant difference between the mean number of miles run per week by group runners and individual runners.

To test if there is a significant difference between the mean number of miles run each week by group runners and individual runners who are training for marathons, we can use a two-sample t-test.

Let's assume that the population standard deviation for the individual runners is also 4.2 miles per week.

We can set up the hypotheses as follows:

Null hypothesis: The mean number of miles run per week by group runners and individual runners is the same.

Alternative hypothesis: The mean number of miles run per week by group runners and individual runners is different.

Mathematically, we can write:

H0: μ1 = μ2

Ha: μ1 ≠ μ2

where μ1 is the population mean for group runners and μ2 is the population mean for individual runners.

We are given the sample mean for the group runners, which is 49.0 miles per week. We do not know the sample mean for the individual runners. Let's assume that we collect a random sample of 32 individual runners and find that their sample mean is 52.0 miles per week.

We can calculate the test statistic as follows:

t = (X1 - X2) / sqrt((s1^2/n1) + (s2^2/n2))

where X1 and X2 are the sample means for group runners and individual runners, s1 and s2 are the population standard deviations for group runners and individual runners, and n1 and n2 are the sample sizes for group runners and individual runners.

Plugging in the values, we get:

t = (49.0 - 52.0) / sqrt((4.2^2/32) + (4.2^2/32))

t = -3.0 / 1.182

t = -2.54

Using a t-distribution table with 62 degrees of freedom (32 + 32 - 2), we can find that the critical value for a two-tailed test with a significance level of 0.05 is approximately ±2.0. Since our calculated t-value (-2.54) is beyond this critical value, we can reject the null hypothesis and conclude that there is a significant difference between the mean number of miles run per week by group runners and individual runners.

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The expression for the area of the triangle is given as follows:

A = 6x² - 7x - 3.

The area of a rectangle of base b and height h is given by half the multiplication of dimensions, as follows:

A = 0.5bh.

The dimensions for this problem are given as follows:

Hence the expression for the area of the triangle is given as follows:

A = 0.5(4x - 6)(3x + 1)

A = 0.5(12x² - 14x - 6)

A = 6x² - 7x - 3.

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[tex]\sqrt[4]{81} -8( \sqrt[3]{216}) +15( \sqrt[5]{32}) +\sqrt{225}[/tex] when simplified gives 0

Indices are small number that tells us how many times a term has been multiplied by itself. Indices are also the power or exponent which is raised to a number or a variable.

How to determine this

[tex]\sqrt[4]{81} -8( \sqrt[3]{216}) +15( \sqrt[5]{32}) +\sqrt{225}[/tex]

When all of then are perfect square

[tex]\sqrt[4]{81}[/tex]= 3 * 3 *3 *3

[tex]\sqrt[3]{216}[/tex] = 6 * 6 * 6

[tex]\sqrt[2]{32}[/tex] = 2 * 2 * 2 * 2 * 2

[tex]\sqrt{225}[/tex] = 15 * 15

Therefore,

3 - 8(6) + 15(2) + 15

3 - 48 + 30 + 15

By collecting like terms

3 + 30 + 15 - 48

48 - 48

= 0

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Connie's team lost 4 out of 16 games this season. To calculate the percentage of games lost, we can divide the number of games lost by the total number of games and then multiply by 100.

In this case, 4 divided by 16 is equal to 0.25, or 25% as a percentage. Therefore, Connie's team lost 25% of their games this season.

We first note that the total number of games played is 16. Connie's team won 12 of these games, so the number of games they lost is 16 - 12 = 4. To calculate the percentage of games lost, we divide the number of games lost by the total number of games and then multiply by 100. This gives:

(4 / 16) * 100 = 25%

Therefore, Connie's team lost 25% of their games this season.

It is important to understand how to calculate percentages in order to solve problems like this. In this case, we used the formula:

percentage = (part / whole) * 100

where the "part" is the number of games lost and the "whole" is the total number of games played. By substituting the appropriate values into this formula, we were able to calculate the percentage of games lost by Connie's team.

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It will take 15 months for the number of basic subscriptions to match the number of deluxe subscriptions.

Let's denote the number of months passed by "m".

In m months, the number of basic subscriptions will be 288 - 5m (since 5 basic subscriptions are decreasing each month), and the number of deluxe subscriptions will be 93 + 8m (since 8 deluxe subscriptions are increasing each month).

We want to find out when the number of basic subscriptions will match the number of deluxe subscriptions, so we can set the two expressions equal to each other:

288 - 5m = 93 + 8m

We can then solve for m:

288 - 93 = 8m + 5m

195 = 13m

m = 15

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Answer: 31

Step-by-step explanation:

We know that the perimeter is equal to 2W + 2L.

       178 cm = 2W + 2L

We also know that the length is 4 cm less than twice the width.

       L = 2W - 4

We will create a system of equations. Then we will solve with substitution.

       178 cm = 2W + 2L

       L = 2W - 4

       178 cm = 2W + 2L

       178 cm = 2W + 2(2W - 4 cm)

       178 cm = 2W + 4W - 8 cm

       178 cm = 6W - 8 cm

       186 cm = 6W

               W = 31

Answer:

31 cm

Step-by-step explanation:

First, we choose two variables.

Let W = width.

Let L = length.

The perimeter of a rectangle is:

perimeter = 2(L + W)

Now we translate the statements of the problem into two equations.

"the perimeter is 178 cm"

2(L + W) = 178

Divide both sides by 2:

L + W = 89

"the length of a rectangle is 4 cm less than the twice of the width"

L = 2W - 4

We have a system of equations:

L + W = 89

L = 2W - 4

Rewrite the first equation:

L + W = 89

Since the second equation is already solved for L, we can easily use the substitution method.

Substitute 2W - 4 for L in the equation above.

2W - 4 + W = 89

Combine like terms on the left side.

3W - 4 = 89

Add 4 to both sides.

3W = 93

Divide both sides by 3.

W = 31

Answer: The width is 31 cm

Check:

The length is 4 cm less than twice the width. 2 × 31 cm - 4 cm = 58 cm

The length is 58 cm, and the width is 31 cm. Calculate the perimeter.

P = 2(L + W) = 2(58 cm + 31 cm) = 2(89 cm) = 178 cm

The perimeter is 178 cm as the problem states, so the answer, width = 31 cm, is correct.

The weight of the goods on the ship decreased by 50 percent.

To find the percentage decrease in the weight of goods on the ship, we need to calculate the difference between the initial weight and the final weight, divide it by the initial weight, and then multiply by 100 to get the percentage decrease.

The initial weight of the goods on the ship was 60,010 tons and the final weight after unloading was 30,005 tons.

The difference between the initial weight and the final weight is 60,010 - 30,005 = 30,005 tons.

To find the percentage decrease, we divide the difference by the initial weight:

30,005 / 60,010 = 0.5

Multiplying by 100 gives us the percentage:

0.5 x 100 = 50%

Therefore, the weight of goods on the ship has decreased by 50 percent.

In conclusion, the percentage decrease in the weight of goods on the ship is 50 percent. This means that the ship has unloaded half of its initial weight and now carries only half of the weight it carried when it left the port last week. This calculation can be helpful for the shipping company to determine the efficiency of their transportation and to plan for future shipments.

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The area of the grass seed needs to cover is 38.625 m².

What is the area of the rectangle?

A quadrilateral with parallel sides that are equal to one another and four equal vertices is known as a rectangle. It is also known as an equiangular quadrilateral for this reason. Rectangles can also be referred to as parallelograms because their opposite sides are equal and parallel.

Here, we have

Given: Tim wants to buy grass seed to cover his whole lawn, except for the pool. The pool is 5 3/4 m by 3 1/2 m.

We have to find the area the grass seed needs to cover.

Let the area of the grass seed needs to cover be A

Now , the area of the lawn L = ( 11 3/4 ) x 5 m²

The area of the lawn L = 58.75 m²

where the length of the pool l= ( 5 3/4 ) m = 5.75 m

The width of the pool is w = ( 3 1/2 ) m = 3.5 m

Now, Area of Rectangle = Length x Width

On simplifying, we get

Area of the pool P = 5.75 x 3.5 = 20.125m²

Now, the area of the grass seed needs to cover A = L - P

The area of the grass seed needs to cover A = 58.75 m² - 20.125 m²

The area of the grass seed needs to cover A = 38.625 m²

Hence, the area of the grass seed needs to cover is 38.625 m².

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Fill the  two-way frequency table as shown in the image attached.

A two-way frequency table is a way of displaying frequencies for two different categories collected from a single group of people.  One category is represented by the rows and the other is represented by the columns.

For the frequency table:

Total = 82

Apple

Total = 24

Students = 22

Teachers = 24 - 22 = 2

Grape

Total = 82 - 25 -24 = 33

Students = 33 - 4 = 29

Orange

Teachers = 25 - 24 = 1

Students

Total  = 22 + 29 + 24 = 75

Teachers

Total  = 2 + 4 + 1 = 7

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It would take 15 years of investing $200 at 8% interest to reach a value of $248.

To solve for t, we can rearrange the formula:

a = p + prt

a - p = prt

t = (a - p) / (pr)

To solve for how long $200 must be invested at 8% interest to reach a value of $248, we can plug in the given values into the formula and solve for t:

a = 248

p = 200

r = 0.08

t = (a - p) / (pr)

t = (248 - 200) / (200 * 0.08)

t = 15

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Answer:

Step-by-step explanation:

You want the median of the Blue box plot and the IQR of the Pink box plot shown in the figure.

A box plot has 5 vertical lines. From left to right, they are ...

Based on the above, we can see the median is the middle line inside the box of the box plot.

The median of the blue plot is 5.2.

The "Interquartile range" is the difference between the 3rd quartile and the 1st quartile. That is, it is the difference between the two ends of the box, the length of the box.

The IQR of the pink plot is 6.4 -5.6 = 0.8.

__

Additional comment

This is basically an exercise in understanding the terms associated with a box plot, and reading a graph. The only calculation involved is figuring out the missing graph coordinates and finding the difference of the coordinates at the ends of the pink plot.

Correspondence is a relation of connection as it establishes a link between two sets of elements, often by relating each element in one set to a specific element in the other set.

Correspondence refers to the exchange of communication or information between two or more parties. It is a relation of connection because it involves establishing a link or connection between the sender and the receiver of the message.

For example, when two people exchange letters or emails, they establish a correspondence that connects them and allows them to communicate. Similarly, in business, correspondence can refer to the exchange of official documents such as letters, memos, and reports, which establish a connection between different departments or organizations. Overall, correspondence is an important aspect of communication that helps to establish and maintain relationships between individuals and groups.

For example, in mathematics, a correspondence can be seen when matching the elements of one set to another, such as associating students with their grades. In this case, the connection is created by linking each student to their respective grade, illustrating the concept of correspondence.

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The quadratic function that models the number of cases of flu each year, where y is years since 2012 is y = -0.02x^2 + 0.5x + 10. The coefficient of x is 0.5.

Suppose the number of cases of flu each year initially increases rapidly, but then starts to level off and eventually decline. We can model this behavior with a quadratic function of the form:

y = ax^2 + bx + c

where y is the number of cases of flu, and x is the number of years since 2012. Estimate the coefficients a, b, and c.

Assume the number of cases of flu was initially very low in 2012, so the y-intercept c is small value, say 10.

Next, assume that the number of cases of flu initially increased rapidly, but then started to level off around 2018.

y = ax^2 + bx + 10

where a is negative and b is positive.

Suppose the coefficient of the linear term is small, since we expect the trend to level off rather than continue to increase at a constant rate.

So, a possible quadratic function that models the number of cases of flu each year is:

y = -0.02x^2 + 0.5x + 10

The coefficient of x in this function is 0.5, which represents the rate of change of the number of cases of flu each year after 2012.

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On the basis of given data & Using combining-functions, We can say that

a).The function P(x) which represents the price of a ticket can we given by where x represents the number of 0.75 price decreases= ($15 - 0.75x)

b). the function T(x) which represents number of tickets sold on a day

=(180 + 10x)

c). the revenue would be if the magician decides to sell the tickets for

$12 =  $ 2640

The process of composition, in which the result of one function becomes the input of another, allows us to create complex functions from basic ones. A complicated function may occasionally need to be broken down into two or more simpler ones, using the opposite method.

Given that:

Current price of ticket = $15

Average tickets sold per day = 180

Price decrease each time = $0.75

Number of price decreases = x

Total Price decrease = 0.75x

Increase in number of tickets with price decrease each time = 10

Total increase in number of tickets with x times price decrease = 10x

a). Function to represent total price of tickets

P(x) = current price - total price decrease  

      = ($15 - 0.75x)

Function to represent number of tickets T(x)

= average number of tickets+ total increase in tickets

= (180 + 10x)

b)Function to represent total revenue R(x) = total tickets x total price

                                                                      = T(x) . P(x)

                                                                      = (180 + 10x) ($15 - 0.75x)

c)Given that total price of tickets=$12

                                            P(x)=$12

                             ($15 - 0.75x) = $12

                                      - 0.75x  =  $12 - $15

                                       - 0.75x  =  - $3

                                              x = 3 ÷ 0.75

                                              x = 4 times

The number of times price decreased=4

Total tickets sold T(x) =(180 + 10x)

                                  =(180 + 10(4))

                                  =180+40

                                   =220 tickets

Total revenue= T(x) . P(x)

                    = 220 x 12

                    =$2640

Total revenue earned on 220 tickets at $12 is $ 2640

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The volume, in cubic feet, of the right rectangular prism is 9 cubic feet

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

A right rectangular prism has a length of 2 feet, a width of 3 feet, and a height of 1 2/4 feet

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

Volume = Length * Width * height

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

Volume = 2 * 3 * (1 2/4)

Evaluate

Volume = 9

Hence, the volume is 9 cubic feet

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The number of people who did not like pop songs is 175.

Given data :

In a survey, the ratio of people who liked pop songs and rap songs is 7: 9.

The number of people who liked both songs = 60.

The number of people who liked only rap songs =  30.

The number of people who liked none of the songs = 40

First of all, we will find the number of people who only like pop songs. We are given a ratio of 7:9 for the people who liked pop songs and rap songs. Let us assume that the number of people who liked pop songs is x and those who liked rap songs is y. According to the ratio given,

[tex]\frac{x}{y} = \frac{7}{9}[/tex]   .....(1)

As we know the number of people who only liked rap songs are 30. Therefore, y - x = 30

x = y - 30

We will substitute the value of x in equation 1.

[tex]\frac{y - 30}{y} = \frac{7}{9}[/tex]

9y - 270 = 7y

2y = 270

y = 135

Now, x = 135 -30

x = 105

Total number of people in survey = x + y + 40

105 + 135 + 40 = 280

Out of 280, the number of people who liked pop songs is 105. So, the number of those who did not like pop songs is ( 280 - 105) = 175.

Therefore, the number of people who did not like pop songs are 175.

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The calculated value of the actual area of the park in meters squared is 8100x

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

A scale drawing of a rectangular park had a scale of 1 cm = 90 m.

This means that

Scale factor = 90/1

Evaluate

Scale factor = 90

The actual area of the park in meters squared is calculated as

Area = Area of scale * Scale factor^2

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

Area = Area of scale * 90^2

Evaluate

Area = Area of scale * 8100

Let Area of scale = x

So, we have

Area = 8100x

Hence, the actual area of the park in meters squared is 8100x

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The confidence level for length of the thorax of 49 fruit fly males is  (0.7820 mm, 0.8188 mm).

Here the sample size (n=49), the sample mean (a=0.8004 mm), and the sample standard deviation (s=0.0782 mm), we can calculate the confidence interval as follows:

1. Determine the critical value (z-score) for a 90% confidence level. You can use a z-table or calculator to find the z-score, which is approximately 1.645.

2. Calculate the margin of error using the z-score, sample standard deviation, and sample size:
Margin of Error = z-score * (s / √n)
Margin of Error = 1.645 * (0.0782 / √49)
Margin of Error ≈ 0.0184 mm

3. Add and subtract the margin of error from the sample mean to obtain the confidence interval:
Lower Limit = a - Margin of Error = 0.8004 - 0.0184 ≈ 0.7820 mm
Upper Limit = a + Margin of Error = 0.8004 + 0.0184 ≈ 0.8188 mm

So, the 90% confidence interval for the true mean thorax length of a male fruit fly is approximately (0.7820 mm, 0.8188 mm). This means that we are 90% confident that the true mean thorax length of a male fruit fly falls within this range.

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The angle θ lies in Quadrant I given that cos(θ) > 0 and sin(θ) > 0.

Based on the given information that cos(θ) > 0 and sin(θ) > 0, we can determine the quadrant in which the angle θ lies.

Recall that there are four quadrants in a Cartesian coordinate system: Quadrant I (both x and y are positive), Quadrant II (x is negative, y is positive), Quadrant III (both x and y are negative), and Quadrant IV (x is positive, y is negative). The cosine function, cos(θ), represents the x-coordinate of a point on the unit circle, while the sine function, sin(θ), represents the y-coordinate.

Since cos(θ) > 0, the angle θ must be in a quadrant where the x-coordinate is positive. This means that θ can lie in either Quadrant I or Quadrant IV. Next, since sin(θ) > 0, the angle θ must be in a quadrant where the y-coordinate is positive. This narrows down the possibilities to only Quadrant I, where both x and y coordinates are positive.

Therefore, the angle θ lies in Quadrant I given that cos(θ) > 0 and sin(θ) > 0.

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