Question

Hercules Films is deciding on the price of the video release of its film Son of Frankenstein. Its marketing people entimate that at a price of p dollars, 2 can sell a total of a a 280,000 - 20 , 00oup eopies What price wil bring in the greatert revenue? p=5

Answer 1

The price that will bring in the greatest revenue for Hercules Films' video release of "Son of Frankenstein" is $10.

Step-by-step explanation:

To maximize revenue, we need to find the price that maximizes the product of the number of units sold and the price per unit. The revenue function is given by
`\( R(p) = p * (280,000 - 20,000p) \).` To find the maximum, we take the derivative of the revenue function with respect to
`\( p \)` and set it equal to zero.

The derivative of the revenue function is
`\( R'(p) = 280,000 - 40,000p \).`Setting this equal to zero and solving for
`\( p \) gives us \( p = 7 \).`However, we also need to check the endpoints of the feasible range, which is
`\( p \geq 0 \). When \( p = 5 \), \( R(p) = 5 * (280,000 - 20,000 * 5) = 5 * 180,000 = 900,000 \). When \( p = 10 \), \( R(p) = 10 * (280,000 - 20,000 * 10) = 10 * 180,000 = 1,800,000 \).`

Therefore, the maximum revenue occurs at
`\( p = 10 \),` and the corresponding price is $10. This is the price that will bring in the greatest revenue for Hercules Films.

Answer 2

The price that will bring in the greatest revenue for Hercules Films' video release of "Son of Frankenstein" is $5.

Step-by-step explanation:

To find the price that maximizes revenue, we need to consider the revenue function, which is the product of the price (p) and the quantity sold (q). In this case, the revenue function is given by R(p) = p * (280,000 - 20,000p). The goal is to maximize R(p) with respect to p. To do this, we can take the derivative of R(p) with respect to p and set it equal to zero to find critical points.

The derivative of R(p) is dR/dp = 280,000 - 40,000p. Setting this equal to zero gives us 280,000 - 40,000p = 0. Solving for p, we find p = 7. The critical point is p = 7, but we also need to check the endpoints of the feasible interval, which is the possible range for p. In this case, the price cannot be negative, so the feasible interval is [0, ∞). Since p = 5 is within this interval, we need to evaluate R(p) at the critical point and endpoints.

R(5) = 5 * (280,000 - 20,000 * 5) = $1,300,000

R(7) = 7 * (280,000 - 20,000 * 7) = $1,260,000

The revenue is highest at p = 5, making $1,300,000 the maximum revenue. Therefore, the optimal price for the video release of "Son of Frankenstein" is $5.