2. [10 marks] Solve the Cauchy problem 2ux + y = cos x = U.2,0) = sina

An assembly time, chain saw requires 5 hours of assembly and a wood chipper 2 hours. From linear programming for maximum profit, number of chain saw assembled are 0 and number of wood chipper are 10.

This problem is related to the topic of linear programming. We use a graphical method to solve the equation. To determine the maximum profit we write the equation of profit. Let we consider x chain saw and y wood chipper. We have a chain saw requires 5 hours of assembly. But wood chipper require 2 hours of assembly. Maximum hours of assembly time available = 20 hours

So, 5x + 2y ≤ 20 --(1) and x ≥ 0, y≥ 0

Profit of chain saw = $190

Profit on wood chipper = $210

Maximize profit, Max Z = 190x + 210y

Now, using graphical method, draw a graph is present in above figure. To determine the profit at point A and B

At point A( 0, 10)

Z = 190 ×(0)+ 210× (10) = 2100 --(1)

At point B(5,0)

Z = 190×(5)+230×(0) = 950 --(2)

Clearly from equation (1), maximum profit is at point A, number of chain saw assembled are 0 and number of wood chipper arre 10.

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The Intermediate Value Theorem states that for any value c between the function's values at the endpoints (f(a) and f(b)), there exists a value x in the interval [a, b] such that f(x) = c.

The assumptions (or conditions) required for the Intermediate Value Theorem are:

1. The function is continuous: The function must be continuous on the closed interval [a, b]. This means that there are no breaks, jumps, or holes in the graph of the function within the given interval.

2. The interval is closed and bounded: The interval [a, b] must be a closed interval, meaning it includes both endpoints a and b. Additionally, the interval must be bounded, meaning the function has a maximum and a minimum value within the interval.

By satisfying these two assumptions, the Intermediate Value Theorem states that for any value c between the function's values at the endpoints (f(a) and f(b)), there exists a value x in the interval [a, b] such that f(x) = c.

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Answer:

√(4.93 × 17.1) is about √(5 × 20) = √100 = 10. 0.209 is about 0.2, or 1/5.

So we have 10/0.2 = 100/2 = 50.

At the end of three years, Daniel would have saved $6749.

When interest is calculated on a principal amount of money using the compound interest method, the interest gained is periodically added back to the principal and interest is then calculated on the new principal amount.

We know that;

A = P(1 + r/n)^nt

A =amount

r = rate

P = principal

n = Number of times compounded

t = time

Thus;

A = 6000(1 + 0.04)^3

A = $6749

Thus the compound amount saved is  $6749

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The  equation of best fit is ŷ = 2.69152X + 3.45818.

From the data we cab write,

Sum of X = 45

Sum of Y = 155.7

Mean X = 4.5

Mean Y = 15.57

Sum of squares (SSX) = 82.5

Sum of products (SP) = 222.05

So, Regression Equation = ŷ = bX + a

Now, b = SP/SSX = 222.05/82.5 = 2.69152

and, a = MY - bMX = 15.57 - (2.69*4.5) = 3.45818

Thus, the equation of best fit is ŷ = 2.69152X + 3.45818.

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Answer:

The graph shown is the answer

Step-by-step explanation:

It says why and how its the answer

The statement is true. A put option is a financial contract that provides the holder with the right, but not the obligation, to sell an underlying asset, such as 100 shares of common stock, at a predetermined price, known as the strike price, before or at the expiration date of the option. If the price of the asset decreases below the strike price, the holder of the put option can sell the asset at the higher strike price and make a profit. However, if the price of the asset increases above the strike price, the holder can simply do nothing and let the option expire, without any obligation to sell the asset.

The price of a put option is influenced by various factors, such as the price of the underlying asset, the strike price, the time until expiration, and the volatility of the asset's price. Generally, the higher the volatility and the longer the time until expiration, the higher the price of the put option.

Selling a put option involves taking on the obligation to buy the underlying asset at the strike price, if the holder decides to exercise the option. Selling put options can be a strategy for generating income, but it also involves risks, such as potential losses if the price of the asset decreases significantly.

In summary, a put option is a security that provides the holder with the right to sell an underlying asset at a predetermined price, but not the obligation to do so. The price of a put option is influenced by various factors, and selling a put option involves taking on the obligation to buy the asset.

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The equation of the line written in general form (4,5) (1,2) is y - x = 1.

The formula to find the line equation from two points is as follows -

(y - [tex] y_{1}[/tex])/(x - [tex] x_{1}[/tex]) = ([tex] y_{2}[/tex] - [tex] y_{1}[/tex])/([tex] x_{2}[/tex] - [tex] x_{1}[/tex]). In the formula, [tex] y_{1}[/tex] and [tex] y_{2}[/tex] are initial and final y-axis values and [tex] x_{1}[/tex] and [tex] x_{2}[/tex] are initial and final x-axis values.

Keep the values in formula to find the equation -

(y - 5)/(x - 4) = (2 - 5)/(1 - 4)

Simplifying the equation

(y - 5)/(x - 4) = -3/-3

Divide the values

(y - 5)/(x - 4) = 1

Rewriting the equation

(y - 5) = (x - 4)

Rearranging the equation

y - x = 5 - 4

y - x = 1

Hence, the equation of line is y - x = 1.

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Answer: 2/3 or 12/18

Step-by-step explanation:

The dimension available for the design is the lateral surface area of the can, which is approximately 251.33 square centimeters.

The area between the bases of the can is the lateral surface area of the can, which can be calculated using the formula:

Lateral surface area = 2 x π x r x h

where π is the mathematical constant pi (approximately 3.14159), r is the radius of the can (which is half of the diameter), and h is the height of the can.

In this case, the height of the can is given as 9.4 centimeters, and the diameter (and hence the radius) is given as 8.5/2 = 4.25 centimeters.

Plugging in these values, we get:

Lateral surface area = 2 x π x 4.25 cm x 9.4 cm

= 251.327 square centimeters

Therefore, the dimension available for the design is the lateral surface area of the can, which is approximately 251.33 square centimeters.

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The average speed of the buses is approximately 19.08 miles per hour.

We have,

If the trip takes 3 hours and 15 minutes of driving time, that's a total of 3.25 hours.

The distance traveled is 124 miles round trip, so the distance traveled in one direction is half of that or 62 miles.

To find the average speed of the buses in miles per hour, we can use the formula:

average speed = distance/time

where distance is in miles and time is in hours.

So, the average speed of the buses in miles per hour is:

Average speed

= 62 miles / 3.25 hours

= 19.0769 miles per hour (rounded to 4 decimal places)

Therefore,

The average speed of the buses is approximately 19.08 miles per hour.

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Answer: m∠1=76.5

Step-by-step explanation:

x+y/2

71+82/2

153/2

76.5

Hope this helps! :)

1. Circling non-ratio.

2. Circling ratios that are fully simplified.

Variables in a proportion:

3. Solve for the variable in each proportion.

a. 3/9 = x/12

To solve for x, cross-multiply and simplify:

3/9 = x/12

(3)(12) = (9)(x)

36 = 9x

x = 4

Therefore, the solution is x = 4.

b. 20/y = 4/3

To solve for y, cross-multiply and simplify:

20/y = 4/3

(4)(y) = (20)(3)

4y = 60

y = 15

Therefore, the solution is y = 15.

4. Solve for the variable in each proportion.

a. 2/14 = (x+1)/28

To solve for x, cross-multiply and simplify:

2/14 = (x+1)/28

(2)(28) = (14)(x+1)

56 = 14x + 14

42 = 14x

x = 3

Therefore, the solution is x = 3.

b. 3/(2y-3) = 6/y

To solve for y, cross-multiply and simplify:

3/(2y-3) = 6/y

(3)(y) = (6)(2y-3)

3y = 12y - 18

-9y = -18

y = 2

Therefore, the solution is y = 2.

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The probability that a randomly selected point within the circle falls in the

red-shaded triangle is 0.14.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

Area of the circle.

= πr²

= 3.14 x 15 x 15

= 706.5

Area of the shaded triangle.

= 1/2 x base x height
= 1/2 x 12 x 16

= 6 x 16

= 96

Now,

The probability that a randomly selected point within the circle falls in the

red-shaded triangle.

= Area of shaded triangle /Area of the circle

= 96/706.5

= 0.14

Thus,

The probability that a randomly selected point within the circle falls in the

red-shaded triangle is 0.14.

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X1 = U1 + U-1 = U1 (since U-1 = 0);
X2 = U2 + U0 = U2 + 1 (since U0 = y1 - y0 = -1);
X3 = U3 + U1 = U3 + U1;
X4 = U4 + U3 + U2.

Innovations algorithm is a method for computing the values of a time series at each time point using the lag operator and the innovation or error terms. The lag operator, denoted by the symbol B, is used to shift the time index of a time series by one unit. Specifically, B multiplied by the time series X is equivalent to the time series X shifted one unit into the past.

We are given the MA(2) process X, = (1+0,B²)W with W, - WN(0,1), this means that X is a moving average process of order 2, where the current value of X depends on the current and two past innovation terms. The innovation terms U1, U2, U3, ... are uncorrelated random variables with zero mean and unit variance.

i) To use the innovations algorithm to find X1, X2, and X3 in terms of U1, U2, and U3, we first need to express X in terms of B and the innovation terms. Using the definition of the MA(2) process, we have:

Xt = (1 + 0*B^2)Wt + B* (1 + 0*B^2)Wt-1 + B^2* (1 + 0*B^2)Wt-2
= (1 + 0*B^2)Wt + B* (1 + 0*B^2)Wt-1 + B^2* (1 + 0*B^2)Wt-2
= Wt + B*Wt-1 + B^2*Wt-2

Next, we can use the innovations algorithm to express Xt in terms of the innovation terms U1, U2, and U3. The algorithm works as follows:

- Compute the initial values of the time series using the given initial conditions. In this case, we have y(0)=1+0*2, y(1)=0, y(2)=0, and y(h)=0 for h>3.
- For each time point t > 2, compute the innovation term Ut as the difference between the observed value yt and the predicted value based on the past observations up to time t-1. Specifically, we have:
Ut = yt - (1 + 0*B + 0*B^2) yt-1
= yt - yt-1

- Compute the current value of Xt as a linear combination of the past innovation terms Ut-1, Ut-2, and Ut-3, using the coefficients from the MA(2) process. Specifically, we have:
Xt = Ut + 0*Ut-1 + Ut-2
= Ut + Ut-2

Using this algorithm, we can compute the values of X1, X2, and X3 as follows:
- X1 = U1 + U-1 = U1 (since U-1 = 0)
- X2 = U2 + U0 = U2 + 1 (since U0 = y1 - y0 = -1)
- X3 = U3 + U1 = U3 + U1

ii) To find the value of X4, we can use the result from part i to express X4 in terms of the innovation terms U1, U2, U3, and U4. Specifically, we have:

X4 = U4 + U2
= y4 - y3 + y2 - y1
= (1 + 0*B^2) W4 - (1 + 0*B + 0*B^2) W3 + (1 + 0*B^2) W2 - (1 + 0*B + 0*B^2) W1
- (1 + 0*B + 0*B^2) W0
= W4 - W3 + W2 - W1
= U4 + U3 + U2

Therefore, X4 = U4 + U3 + U2.

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Answer: Geometry is a branch of mathematics that deals with the study of shapes, sizes, positions, and measurements of objects in two-dimensional and three-dimensional spaces. It involves analyzing and calculating angles, lengths, areas, volumes, and other properties of various figures such as triangles, circles, squares, cubes, and spheres. Geometry is used in many fields including architecture, engineering, physics, and computer graphics.

Answer:

If not choose all that apply, technological process will shift the PPC right (outwards)

Step-by-step explanation:

The production possibilities curve represents the maximum amount of goods and services that a country can produce with its limited resources. It is a graphical representation of the trade-offs a country must make between producing one good or service over another. Any factor that can increase production will shift the curve outward, indicating an increase in the maximum amount of goods and services that can be produced.

An increase in the production of investment goods will shift the curve outward because it will increase the amount of capital goods that can be used to produce other goods and services. Investment goods are items like machinery, equipment, and infrastructure that are used to produce other goods and services. By increasing the production of investment goods, a country can increase its productive capacity, allowing it to produce more goods and services in the future.

An increase in the production of consumer goods will not shift the curve outward because it does not increase the country's productive capacity. Consumer goods are items that are consumed immediately, like food, clothing, and electronics. While an increase in the production of consumer goods can lead to short-term economic growth, it will not increase the country's long-term productive capacity.

Technological progress, on the other hand, will shift the production possibilities curve outward. Technological progress refers to advancements in technology that increase productivity, reduce costs, and improve efficiency. By adopting new technologies, a country can produce more goods and services with its existing resources, allowing it to shift its production possibilities curve outward.

In conclusion, an increase in the production of investment goods and technological progress will shift the production possibilities curve outward, while an increase in the production of consumer goods will not.

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The equation for the temperature, d, in terms of t is:
d(t) = 9 * cos[(π/12) * (t - 17)] + 80

To create an equation for the temperature, d, in terms of t, we will use the information given: the high temperature of 89 degrees at 5 pm and the average temperature of 80 degrees.

Determine the amplitude (A) of the sinusoidal function.
Amplitude = (High Temperature - Average Temperature)
A = (89 - 80)
A = 9

Determine the period (P) of the sinusoidal function.
Since the temperature pattern repeats every 24 hours, the period is 24 hours.

Determine the horizontal shift (HS) of the sinusoidal function.
Since the high temperature occurs at 5 pm (17 hours since midnight), the horizontal shift is 17 hours.

Determine the vertical shift (VS) of the sinusoidal function.
The vertical shift is the average temperature, which is 80 degrees.

Write the sinusoidal equation for the temperature, d, in terms of t.
Since the temperature reaches its peak (high temperature) at 5 pm, we will use the cosine function, as it starts at its peak value. The general form of the cosine function is:

d(t) = A * cos[(2π/P) * (t - HS)] + VS

Now, plug in the values found in steps 1-4:

d(t) = 9 * cos[(2π/24) * (t - 17)] + 80

So, the equation for the temperature, d, in terms of t is:

d(t) = 9 * cos[(π/12) * (t - 17)] + 80

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Rose sold 53.6 pretzels.

P x ? + W = 135

2.50 x 53.6 + 1.00 = 135

The scatter plot shows that there is no strong correlation between average weight and life expectancy for the 12 dog breeds. The coefficient of correlation for this adult weight data to be -0.123, which is very near to 0.

The adult weight of a person can vary greatly depending on several factors, including genetics, age, gender, height, muscle mass, and overall health. However, a healthy weight range for adults is typically determined by body mass index (BMI), which is calculated by dividing weight in kilograms by height in meters squared.

We can use the average weight as the x-value and the average life expectancy as the y-value to represent the data as ordered pairs. Following are the ordered pairs for the provided data:

(125, 57.5)

(62.5, 49)

(12, 16)

(12.3, 12)

(11, 11)

We may plot these ordered pairs on a scatter plot to show the form of the data. As for the story:

We can observe from the scatter plot that there is no obvious pattern formed by the data points. There is no discernible pattern or connection between the two variables, and the spots are dispersed throughout the plot. As a result, the data's shape can be described as random or dispersed.

We can calculate the correlation coefficient to provide more evidence for this. The variables do not have a linear relationship if the correlation coefficient is close to zero. we can calculate the coefficient of correlation for this data to be -0.123, which is very near to 0.

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A)  The demand function is [tex]y = 500 - 20x[/tex]

B) The revenue function [tex]R(x) = 500x - 20x^2[/tex]

C) The marginal revenue [tex]MR(x) = 500 - 40x[/tex]

D) This means that the store can maximize its revenue by setting the price at $25

E) The price corresponding to the value found in part d) is $25

To determine the demand function, we need to use the information given in the problem that for each $1.50 decrease in price, sales will increase by 5 travel mugs. Let x be the price in dollars and y be the number of travel mugs sold.

A) Then, we can write the demand function as:

[tex]y = 140 + 5((24-x)/1.5)[/tex]

Simplifying this expression, we get:

[tex]y = 140 + 20(16 - x)[/tex]

[tex]y = 500 - 20x[/tex]

Therefore, the demand function is[tex]y = 500 - 20x.[/tex]

B) The revenue function is given by the product of the price and the quantity sold:

[tex]R(x) = x(500 - 20x)[/tex]

[tex]R(x) = 500x - 20x^2[/tex]

C) The marginal revenue is the derivative of the revenue function with respect to x:

[tex]MR(x) = dR/dx[/tex]

[tex]= 500 - 40x[/tex]

D) To solve for R(x) = 0, we need to set the revenue function equal to zero and solve for x:

[tex]500x - 20x^2 = 0[/tex]

[tex]x(500 - 20x) = 0[/tex]

[tex]x = 0[/tex] or [tex]x = 25[/tex]

Since the price cannot be zero, the store can only set the price at $25 to achieve zero revenue. This means that the store can maximize its revenue by setting the price at $25.

E) The price corresponding to the value found in part d) is $25.

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Answer:The answer is $911.68.

Step-by-step explanation:

Answer:

we will need 20 glass panels to cover the walls of the greenhouse with 5' x 4' glass panels.

Step-by-step explanation:

First, let's calculate the total wall area:

2 walls (10' x 8') = 160 sq. ft.

2 walls (15' x 8') = 240 sq. ft.

Total wall area = 400 sq. ft.

Next, let's calculate the total floor area:

15' x 5' = 75 sq. ft. (total floor area)

75 sq. ft. / 2 = 37.5 sq. ft. (floor area for tomatoes)

To cover the walls with glass panels, we need to divide the total wall area by the area of each glass panel:

Glass panel area = 5' x 4' = 20 sq. ft.

400 sq. ft. (total wall area) / 20 sq. ft. (area of each glass panel) = 20 glass panels

Therefore, we will need 20 glass panels to cover the walls of the greenhouse with 5' x 4' glass panels.

To reflect a figure over the y-axis, we simply change the sign of its x-coordinates while keeping the y-coordinates the same. So, the coordinates of the vertices after a reflection over the y-axis would be:

A'(-2, 1), B'(-1, 3), C'(1, 2), D'(2, 0)

When we reflect a figure over the y-axis, we are essentially flipping the figure horizontally along a vertical line passing through the origin.

This means that every point on the original figure is reflected across this vertical line, resulting in a new figure that is a mirror image of the original.

Thus, the coordinates of the vertices after a reflection over the y-axis are A'(-2, 1), B'(-1, 3), C'(1, 2), and D'(2, 0).

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From percentage formula, in a vine gunshot sound effect download unblocked restaurants, the number of expected Italian restaurants in next 24 menus slipped under Eli's door is six.

We have a Vine gunshot sound effect download unblocked Restaurants often slip takeout menus. Total collected menus by Eli = 28

In this 28 menus, 7 for Italian food. We have to determine the how many menus in next 24 menus slipped under Eli's door expected from Italian restaurants. Percentage is calculated by dividing the value by the total value, and then multiplying the resultant value by 100.

Using percentage formula, 7 are Italian food out of all 28 that is total [tex]\frac{7 {28} × 100 = 25\%[/tex].

Let the required number of menus slipped under Eli's door should you expect to be from Italian restaurants be x. So, number of menus slipped under Eli's door should you expect to be from Italian restaurants out of 24, x = 25% of 24

=> [tex]\frac{25}{100} × 24 = x[/tex]

=> x = 6

Hence, required value is 6.

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The area of the sector is  305. 2 cm²

The area of the shaded part is equivalent to the area of the sector.

The formula for determining the area of a sector is expressed with the equation;

A = θ/360 × πr²

Given that the parameters of the equation are;

From the diagram shown, we have;

Area = 120/360 × 3.14 × 18²

find the square values

Area = 0. 3 × 3. 14 × 324

multiply the values

Area = 305. 2 cm²

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The president of Doerman Distributors, Inc., believes that 27% of the firm's orders come from first-time cu:

sample of 100 orders will be used to estimate the proportion of first-time customers. Use z-table.

0.30. What is the sampling distribution of p for this study?

a. Assume that the president is correct and p

=

Hint(s) Che

A normal distribution because np and n(1-p) are both greater than 5 v

b. What is the probability that the sample proportion will be between 0.20 and 0.40 (to 4 decimals)?

c. What is the probability that the sample proportion will be between 0.25 and 0.35 (to 4 decimals)?

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Partially Correct

Answer:

Step-by-step explanation:

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a. Since the sample size is large enough (n=100) and the expected number of first-time customers (np=27) and the expected number of repeating customers (n(1-p)=73) are both greater than 5, we can assume that the sampling distribution of the sample proportion p is approximately normal with mean μ=p=0.27 and standard deviation σ=sqrt[p(1-p)/n]=sqrt[(0.27)(0.73)/100]=0.046.

b. To find the probability that the sample proportion will be between 0.20 and 0.40, we need to standardize the values using the formula z = (x - μ) / σ, where x is the sample proportion, μ is the population proportion, and σ is the standard deviation of the sampling distribution of p. Then we can use a standard normal distribution table (z-table) to find the corresponding probabilities.

z(0.20) = (0.20 - 0.27) / 0.046 = -1.52

z(0.40) = (0.40 - 0.27) / 0.046 = 2.83

Using the z-table, the probability of getting a z-score between -1.52 and 2.83 is approximately 0.9747. Therefore, the probability that the sample proportion will be between 0.20 and 0.40 is 0.9747 (to 4 decimals).

c. To find the probability that the sample proportion will be between 0.25 and 0.35, we can follow the same steps as in part b:

z(0.25) = (0.25 - 0.27) / 0.046 = -0.43

z(0.35) = (0.35 - 0.27) / 0.046 = 1.74

Using the z-table, the probability of getting a z-score between -0.43 and 1.74 is approximately 0.5267. Therefore, the probability that the sample proportion will be between 0.25 and 0.35 is 0.5267 (to 4 decimals).

The true population proportion and the sample proportion, given the confidence level and sample size.

In statistics, P-hat represents the sample proportion and E represents the margin of error.

Given a confidence interval of (0.444, 0.484), we can find P-hat and E as follows:

P-hat = (lower limit + upper limit) / 2

P-hat = (0.444 + 0.484) / 2

P-hat = 0.464

Therefore, the sample proportion (P-hat) is 0.464.

To find E (the margin of error), we need to use the formula:

E = (upper limit - lower limit) / 2

E = (0.484 - 0.444) / 2

E = 0.02

Therefore, the margin of error (E) is 0.02.

Note that the margin of error indicates the maximum likely difference between the true population proportion and the sample proportion, given the confidence level and sample size.

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The average rate of change of the given function graph between the two given values is: 6

The Average Rate of Change of a function is one that is defined as the average rate at which one quantity is changing with respect to another thing changing. Thus, an average rate of change function is basically a process that helps to calculate the amount of change in one item divided by the corresponding amount of change in another.

It is given by the formula:

f'(x) = [f(b) - f(a)]/(b - a)

From the graph attached, we have that:

f(-4) = -6

f(-2) = 6

Thus:

f'(x) = (6 - (-6))/(-2 - (-4))

f'(x) = (6 + 6)/(-2 + 4)

f'(x) = 6

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Answer:  A

Step-by-step explanation:

Who are more likely to respond to an online poll about the Loch Ness monster may not be representative of the general population, and may have different beliefs about its existence.

The correct answer is: H0: p <= 0.5, H1: p > 0.5

Explanation:

The null hypothesis (H0) is that the proportion of people who believe that the Loch Ness monster exists is less than or equal to 0.5. The alternative hypothesis (H1) is that the proportion is greater than 0.5.

This is a one-tailed test, since we are only interested in whether the proportion is greater than 0.5.

The fact that Internet users who saw the question could decide whether to respond may affect the conclusion of the test, as it introduces potential bias into the sample. Those who are more likely to respond to an online poll about the Loch Ness monster may not be representative of the general population, and may have different beliefs about its existence.

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