Question

Manufacturer finds that the profit EP from assembling r bicycles per day is given by P(x) = -2x^2 + 502x - 200.

a. How many bicycles should be assembled per day to maximize the profit?
b. Find the maximum profit.
c. What is the loss made if no bicycles are assembled in a day? Suggest why this loss would be made.

Answer 1

Maximizing profit for bicycle assembly requires producing 125 bicycles per day, leading to a maximum profit that can be found by substituting x = 125 into the quadratic profit function. A loss occurs when no bicycles are produced, likely due to covering fixed costs.

Step-by-step explanation:

Optimizing Profit for Bicycle Production

To find the number of bicycles that should be assembled per day to maximize profit, we can examine the profit function P(x) = -2x2 + 502x - 200. The vertex form of a quadratic equation gives the maximum or minimum point, and since the coefficient of x2 is negative, this function has a maximum profit. The vertex x-coordinate, which gives the number of bicycles, is found by -b/(2a), where a is the coefficient of x2 and b is the coefficient of x. Therefore, the number of bicycles for maximum profit is -502/(2*(-2)) = 502/4 = 125.5, which means 125 bicycles (since we can't produce half a bicycle) should be assembled.

To calculate the maximum profit, substitute x = 125 into the profit function:
P(125) = -2(125)2 + 502(125) - 200, which delivers the maximum profit.

If no bicycles are assembled, then the loss made would be P(0) = -200. This loss occurs due to fixed costs which must be covered regardless of production levels, such as rent, utilities, or salaried employees.