Question

Find the rate of change of total​ revenue, cost, and profit with respect to time. Assume that​ R(x) and​ C(x) are in dollars. ​R(x)equals60 x minus 0.5 x squared​, ​C(x)equals3 x plus 5​, when xequals40 and dx divided by dtequals15 units per day

Answer 1

Explanation:

Given the Revenue in dollars modelled by the function R(x) = 60x-0.5x²

Cost in dollars C(x) = 3x+5

Profit function = Revenue - Cost

P(x) = R(x) - C(x)

P(x) = 60x-0.5x²-(3x+5)

P(x) = 60x-0.5x²-3x-5

P(x) = -0.5x²+57x-5

The rate of change of total revenue = dR(x)/dt

dR(x)/dt = dR(x)/dx * dx/dt

dR(x)/dx = 60-2(0.5)x²⁻¹

dR(x)/dx = 60-x

Given x = 40 and dr/dx = 15 units per day

dR(x)/dt = (60-x)dx/dt

dR(x)/dt = (60-40)*15

dR(x)/dt = 20*15

dR(x)/dt = 300dollars

Rate of change of revenue = 300dollars

For the rate of change of cost;

dC(x)/dt = dC(x)/dx * dx/dt

dC(x)/dt = 3dx/dt

dC(x)/dt when dx/dt = 15 will give;

dC(x)/dt = 3*15

dC(x)/dt = 45 dollars.

Rate of change of revenue = 45dollars

For the profit;

Profit = Rate of change of revenue - rate of change of cost

Profit made = 300-45

profit made = 255 dollars