Question

Find the number of units that must be produced and sold in order to yield the maximum profit, given the following profit, given the following equations for revenue and cost: R(x)=20x−0.5x² C(x)=5x+4.

Answer 1

Here's a detailed answer using calculus.

We are given the revenue function R(x)=20x−0.5x² and cost function C(x)=5x+4, where x is the number of units produced and sold.

First, we need to determine the profit function P(x). Profit is calculated as the difference between total revenue and total cost.

So, the profit function P(x) = R(x) - C(x), which gives us P(x) = (20x−0.5x²) - (5x+4). Simplifying this expression, we obtain the profit function as P(x) = 15x - 0.5x² - 4.

To maximize profit, we need to find the derivative of the profit function with respect to 'x'. This derivative function will give us the rate at which profit is changing with respect to the number of units produced and sold.

So, we take the derivative of P(x) to obtain P'(x), which comes out to be P'(x) = 15 - 1.0x.

Setting this derivative equal to zero will give us the critical points of the profit function, which are candidates for local maximum and minimum profit values. We set P'(x) = 0, which gives us the equation 15 - 1.0x = 0.
Solving this equation will give x = 15.

Therefore, to maximize profit, 15 units should be produced and sold.