Question

Economy problem. A factory presents the following functions of production cost and income from sales based on the units produced (x). Income I (x) = 25x + 550 Cost C (x) = 0.40x2 - 20x + 1000 The profit function is obtained by subtracting income less cost, G (x) = I (x) - C (x) Find the quantity x to produce, with which the maximum gain is achieved. Note: Calculate G (x) and look for the axis of symmetry. Explain each step.

Answer 1

The quantity x which produces the maximum income is x = 56.25

if we want to find the quantity x such that it produces the maximum gain, firstly we need to reduce the profit function and then take the derivative of that function and find its roots

Since we know that

I(x) = 25x + 550

C(x) = 0.4x² - 20x + 1.000 ⇒-C(x) = -0.4x² + 20x - 1.000

G(x) = I(x) - C(x) = 25x + 550 -0.4x² + 20x - 1.000 = -0.4x² + 45x - 450

G(x)=  -0.4x² + 45x - 450

Now, if we take its derivative

G'(x) = 2(-0.4)x + 45 = -0.8x + 45

To find the roots of that function, we simply find the values that makes the function equals to 0. Thus

-0.8x + 45 = 0

45 = 0.8x

x = 45/0.8 = 450/8 = 225/4 = 56.25

It means that the quantity x which produces the maximum income is x = 56.25