Questions: A local club is arranging a charter flight to Hawaii. The cost of the trip is 583 each for 83 passengers, with a refund of 5 per passenger for each passenger in excess of 83. a. Find the number of passengers that will maximize the revenue received from the flight. b. Find the maximum revenue. a. The number of passengers that will maximize the revenue received from the flight is (Round to the nearest integer as needed.)

A local club is arranging a charter flight to Hawaii. The cost of the trip is 583 each for 83 passengers, with a refund of 5 per passenger for each passenger in excess of 83.
a. Find the number of passengers that will maximize the revenue received from the flight.
b. Find the maximum revenue.
a. The number of passengers that will maximize the revenue received from the flight is (Round to the nearest integer as needed.)

Transcript text: A local club is arranging a charter flight to Hawaii. The cost of the trip is $\$ 583$ each for 83 passengers, with a refund of $\$ 5$ per passenger for each passenger in excess of 83. a. Find the number of passengers that will maximize the revenue received from the flight. b. Find the maximum revenue. a. The number of passengers that will maximize the revenue received from the flight is (Round to the nearest integer as needed.) $\square$

Solution

Solution Steps

Step 1: Revenue Function Construction

The revenue function is constructed based on the given parameters as follows: $$Revenue = n \times (P - R \times (n - M))$$

Step 2: Differentiation and Critical Points

Differentiating the revenue function with respect to $n$ and setting it to zero gives: $$0 = P - 2Rn + RM$$ Solving for $n$ gives the number of passengers that maximizes revenue: $$n = \frac{P + RM}{2R} = 99.8$$

Step 3: Second Derivative Test

The second derivative of the revenue function with respect to $n$ is always negative: $$\frac{{d^2(Revenue)}}{{dn^2}} = -2R$$ This confirms that the critical point is a maximum.

Step 4: Calculate Maximum Revenue

Substituting the value of $n$ that maximizes revenue back into the original revenue function: Maximum Revenue = 49800.2