Question

The daily profit, P, of an oil refinery is given by P = 8x − 0.02x2, where x is the number of barrels of oil refined. How many barrels will give maximum profit and what is the maximum profit?

Answer 1

The maximum profit occurs when the number of barrels are 200.

The maximum profit is 800.

Step-by-step explanation:

The expression for daily profit is given as:

$P=8x-0.02x^2$

In order to maximise profit, the derivative of it must be 0 because for a function, the maximum or minimum occurs when the first derivative is 0.

Therefore, $\frac{dP}{dx}=0$

$\frac{d}{dx}(8x-0.02x^2)=0\\8\frac{dx}{dx}-0.02\frac{dx^2}{dx}=0\\8-0.02(2x)=0\\8-0.04x=0\\0.04x=8\\x=\frac{8}{0.04}=200$

Therefore, the maximum profit occurs when the number of barrels are 200.

The maximum profit when $x=200$ is given as:

$P_{m}=8(200)-0.02(200)^2\\P_m=1600-0.02\times 40000\\P_m=1600-800=800$

Therefore, the maximum profit is 800.