Question

Suppose a product's revenue function is given by (R(x) = 12x - 0.02x^2), where (x) is in units sold. Also, its cost function is given by (C(x) = 8x + 0.005x^2), where (x) is in units produced. Find the maxiμm profits possible. Note: If necessary, round the quantity that maximizes profit to the nearest whole value and use that to compute profits.

a) 150 units; $1,800 profit

b) 200 units; $1,600 profit

c) 250 units; $1,500 profit

d) 300 units; $1,200 profit

Answer 1

The quantity of output that will provide the highest level of profit can be found by determining the vertex of the profit function.

Step-by-step explanation:

Based on the given revenue and cost functions, the maximum profit can be calculated by finding the quantity of output where the difference between total revenue and total cost is the largest. This can be done by determining the vertex of the profit function.

The profit function can be found by subtracting the cost function from the revenue function: P(x) = R(x) - C(x). In this case, P(x) = (12x - 0.02x^2) - (8x + 0.005x^2).

To find the vertex, we can rewrite the profit function as a quadratic equation in standard form: P(x) = -0.025x^2 + 4x.

The x-coordinate of the vertex is given by x = -b/(2a), where a = -0.025 and b = 4. Calculating the value of x gives x = -4/(2*(-0.025)) = 80.

Therefore, the quantity of output that will provide the highest level of profit is 80 units.