Calculus Question

Question

Given the revenue and cost functions R=32x-0.8x^2 and C=5x+12, where x is the daily production, find the rate of change of profit with respect to time when 15 units are produced and the rate of change of production is 8 units per day

Answer 1

Profit = Revenue - Cost so

P(x) = (32x - 0.8x²) - (5x + 12)

P(x) = -0.8x² + 27x - 12

To find the rate of change per time (per day actually) we can take the derivative with respect to time:

dP/dt = -1.6x•(dx/dt) + 27•(dx/dt)

We are being asked to find "the rate of change of profit with respect to time" which is dP/dt

We are give x = 15 and "the rate of change of production" or dx/dt = 8

So dP/dt = -1.6(15)(8) + 27(8) = 24

The units of profit is $

The units of time is days

So the answer is $24/day

Answer 2

Let P(x) = profit when x units are produced and sold

P(x) = R(x) - C(x) = (32x - 0.8x²) - (5x + 12) = -0.8x² + 27x - 12

Find dP/dt when x = 15 and dx/dt = 8:

By the Chain Rule, dP/dt = (dP/dx)(dx/dt) = (-0.16x + 27)(8) = ( -0.16(15) + 27)(8) = $196.8 / day