Question

A movie studio is deciding on the price of the video release of its most recent film. Its marketing people estimate that at a price of p dollars, it can sell =400−10p thousand copies, but each copy costs $5 to make. What price will give the greatest profit?

Answer 1

To achieve the greatest profit, the movie studio should set the price of the video release at $22.50 per copy. This price is determined by taking the derivative of the profit function and setting it to zero, which shows the point at which profit is maximized before production costs outpace revenue.

Step-by-step explanation:

To determine the price that will yield the greatest profit for the movie studio, we need to set up the profit function. Profit (π) can be calculated by multiplying the number of units sold (400 - 10p) by the price per unit (p) and then subtracting the total cost of production for that number of units, which is $5 per unit. The profit function is therefore π(p) = p(400 - 10p) - 5(400 - 10p). To find the price that maximizes profit, we set the derivative of the profit function π'(p) to zero and solve for p.

The derivative π'(p) = 400 - 20p - 5(-10) simplifies to -20p + 450. Setting this equal to zero gives us 450 = 20p, which means p = 22.5. Therefore, the price that maximizes profit is $22.50.

Moreover, to ensure this price results in a maximum profit rather than a minimum or inflection point, we must check the second derivative of the profit function, π''(p), which should be negative. Indeed, π''(p) = -20, confirming that p = $22.50 does indeed maximize profits.