Question

The president of a toy car manufacturing company believes that P(c) = -2c² + 14c - 20 represents the profit, where c is the number of cars produced (in thousands). Determine the maximum profit the company can earn.

a) $100,000
b) $120,000
c) $140,000
d) $160,000

Answer 1

The maximum profit the company can earn is $4,500.

Step-by-step explanation:

To determine the maximum profit the company can earn, we need to find the vertex of the quadratic function P(c) = -2c² + 14c - 20. The vertex of a quadratic function in the form ax² + bx + c is given by the formula c = -b / (2a). Plugging in the values from the given function, we get c = -14 / (2 * -2) = 7 / 2 = 3.5.

Therefore, the maximum profit the company can earn is obtained when they produce 3.5 thousand cars. To find the maximum profit, we substitute c = 3.5 back into the original function: P(3.5) = -2(3.5)² + 14(3.5) - 20 = -24.5 + 49 - 20 = 4.5.

Hence, the maximum profit the company can earn is $4,500.