Hercules Films is deciding on the price of the video release of its film Bride of the Son of Frankenstein. Marketing estimates that at a price of p dollars, it can sell

q = 240,000 − 15,000p
copies, but each copy costs $4 to make. What price will give the greatest profit?

To determine the price that will give the greatest profit, we need to consider the relationship between the price, the quantity sold, and the cost.

The profit can be calculated as the difference between the revenue and the cost. The revenue is the product of the price and the quantity sold.

Let's break down the problem step by step:

1. Start by calculating the revenue:
Revenue = Price * Quantity sold
R(p) = p * (240,000 - 15,000p)

2. Calculate the cost:
The cost per copy is $4, and the quantity sold is given by the equation q = 240,000 - 15,000p.
Cost = Cost per copy * Quantity sold
C(p) = 4 * (240,000 - 15,000p)

3. Calculate the profit:
Profit = Revenue - Cost
P(p) = R(p) - C(p)
Substitute the expressions for revenue and cost we obtained in steps 1 and 2 into the profit equation.

4. Simplify the profit equation:
P(p) = p * (240,000 - 15,000p) - 4 * (240,000 - 15,000p)
P(p) = 240,000p - 15,000p^2 - 960,000 + 60,000p

5. Rearrange the profit equation to make it easier to work with:
P(p) = -15,000p^2 + 300,000p - 960,000

Now we have the profit equation, P(p), in terms of the price variable p. We can use this equation to find the price that will give the greatest profit.

To find the price that maximizes the profit, we can use techniques from calculus. We need to find the critical points by taking the derivative of the profit function and setting it equal to zero:

dP(p)/dp = -30,000p + 300,000 = 0

Solving this equation, we find:
-30,000p = -300,000
p = 10

Now we need to determine if this critical point is a maximum or minimum by taking the second derivative:

d²P(p)/dp² = -30,000

Since the second derivative is negative, the critical point p = 10 is a maximum.

Therefore, the price that will give the greatest profit is $10.