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# On a certain route, a regional airline carries 8500 per month, each paying $65. The airline wants to increase the fare.

On a certain route, a regional airline carries 8500 per month, each paying $65. The airline wants to increase the fare. However, the market research department estimates that for each $1 in fare the airline will lose 100 passenger. Determine the price that maximizes the airline's revenue.

The last sentence of the problem statement indicates that the revenue is being maximized. Let x be the number of passengers, p be the price they pay, and R represent the amount of revenue generated. Write the objective function in terms of x and p.

R(x)=??

(simplify your answer.)

Next, write the constraint equation that relates the price p to the number of passengers x. How is the constraint equation found? (MULTIPLE CHOICE QUESTION)

A. Use the point-slope formula to define the profit function because the relationship between price and number of passengers is linear

B. Use the point-slope formula because the relationship between the price of a ticket and the number of passengers is linear.

C. Use the point-slope formula because the relationship between the price of a ticket and the number of passengers is parabolic

D. It is given in the problem statement.

(End of multiple choice questions)

Note that (x,p) = (8500,65) and (x,p) = (????,66) are on the demand curve.

Let m be the slope of the point-slope equation. Write m as the ratio of the change in price level and the change in passengers.

m=???

Solve the point-slope function in terms of p, the price of a ticket. Write the equation in the form p = mx + b

p=????

(Simplify your answers. Use integers or fractions for any numbers in the expression. Do not factor.)

Rewrite the objective equation in terms of x.

R = px

R= ??? Substitute the expression for p calculated earlier.

(Simplify your answer. Use integers or fractions for any numbers in the expression. Do not factor.)

What is the next step? (MULTIPLE CHOICE QUESTION)

A. Evaluate R''(0) to find the local maximum/minimum.

B. Solve R'(x) = 0 to find the critical value(s).

C. Solve R''(x) = 0 to find the critical value(s).

D. Evaluate R'(0) to find the local maximum/minimum.

(End of multiple choice question)

Find the first derivative of the revenue function.

R'(x)=?????

(Simplify your answer. Use integers or fractions for any numbers in the expression. Do not factor.)

Now set the derivative equation equal to zero and solve for x.

R'(x) = 0

x=????

Why should the second derivative be found? (MULTIPLE CHOICE QUESTION)

A. To find the optimized ticket price.

B. To find the optimized revenue

C. To find the slope of the constraint equation.

D. To determine if the critical value is a maximum or minimum

(End of multiple choice question)

Find R''

R''=????

(Simplify your answer. Use integers or fractions for any number in the expression. Do not factor.)

Since the value of R''(7500) is (negative?/positive?/zero?), the critical value corresponds to a local (minimum?/maximum?).

Now that the critical value for the amount of passengers is confirmed as x=7500, use the constraint equation p calculated earlier to find the ticket price which maximizes revenue.

p(7500)=$?????

(End of questions)