Question

Profit = Revenue - Cost

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

Given: R(x) = x^2 - 30x

Given: C(x) = 5x + 100

X is hundreds of items sold / P, R, C are in hundreds of dollars

(1) Determine the Initial Cost?

(2) Determine the maximum Profit and number of items required for that profit?

(3) Determine the maximum Revenue and number of items required for that revenue?

(4) Find the break even points, P(x) = 0. What do the values represent in relation to the business?

Answer 1

I would request you to ensure that the R(x) and C(x) are typed in correctly.

If yes, below is the solution to this question.

(1) Determine the Initial Cost?

Initial costs are those that are incurred during the setting up process. Here the cost function is given by

C(x) = 5x + 100

The 5x component is affected by the hundreds of units of x produced/ sold. The $100 component is the initial cost or the cost to set up and is not affected by the units of x produced.

Therefore, $100 is the initial cost

(2) Determine the maximum Profit and number of items required for that profit?

To calculate the maximum profit, we will use calculus.

Profit = Total Revenue - Total Cost

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

P(x) = x^2 - 30x - 5x - 100

=> P(x) = x^2 -35x -100

Differentiating profit w.r.t. to x

P'(x) = 2x - 35

Differentiating the profit equation again

P''(x) = 2

Which is positive. This implies that profit can be maximized at x = infinity

The bigger the value of x, the bigger is the profit

3) Determine the maximum Revenue and number of items required for that revenue?

Similarly the second order diffrential of the revenue function is positive, which implies the higher the number of items of x the higher the revenue

(4) Find the break even points, P(x) = 0. What do the values represent in relation to the business?

for P(x) = 0, x^2 - 35x - 100 = 0

Solving this using quardratic equation formula

x = 37.656 or -2.6556

Assuming x is positive, the value of x = 37.656 for P(x) =0