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 understand the relationship between price, quantity, cost, and profit.
Let's break down the information given:
- The quantity sold, q, is given by the equation: q = 240,000 - 15,000p. This equation represents the demand curve, showing how many copies will be sold at different prices.
- The cost to make each copy is $4. We'll call this cost c.
To calculate profit, we subtract the cost from the revenue. The revenue is the product of the price and the quantity sold (p * q):
Revenue = p * q
= p * (240,000 - 15,000p)
Profit = Revenue - Cost
= (p * (240,000 - 15,000p)) - (4 * q)
= (p * (240,000 - 15,000p)) - (4 * (240,000 - 15,000p))
Now, we need to find the price that maximizes profit. To do this, we can differentiate the profit function with respect to p and set it equal to zero. Then, solve for p to find the critical points.
Differentiating the profit function:
d(Profit)/dp = d((p * (240,000 - 15,000p)) - (4 * (240,000 - 15,000p)))/dp
To simplify, we can distribute and combine like terms:
Profit' = (240,000 - 15,000p) - 15,000(p) - 4
Now, set Profit' equal to zero and solve for p:
0 = (240,000 - 15,000p) - 15,000(p) - 4
Simplifying the equation:
0 = 240,000 - 15,000p - 15,000p - 4
Combine like terms:
0 = 240,000 - 30,000p - 4
0 = 240,000 - 30,000p - 4
0 = 240,000 - 30,000p - 4
0 = 240,000 - 30,000p - 4
Now, solve for p:
240,000 - 4 = 30,000p
236,000 = 30,000p
p = 236,000 / 30,000
p ≈ 7.867
The price that will give the greatest profit is approximately $7.87.