A small tie shop finds that at a sales level of x ties per day its marginal profit is MP (x) dollars per tie, where MP (x) equals 1.85 plus 0.12 x minus 0.0024 x squared. Also, the shop will lose $65 per day at a sales level of x equals 0. Find the profit from operating the shop at a sales level of x ties per day.

Correct question:

A small tie shop finds that at a sales level of x ties per day its marginal profit is MP (x) dollars per tie, where

MP (x) = 1.85 + 0.12x - 0.0024x². Also, the shop will lose $65 per day at a sales level of x = 0. Find the profit from operating the shop at a sales level of x ties per day.

Answer:

P (x) = 1.85x + 0.06x² - 0.0008x³ - 65

Step-by-step explanation:

Given the marginal profit function:

MP (x) = 1.85 + 0.12x - 0.0024x², P (0) = - 65

We are to find P (x).

P (x) = ∫MP (x) dx

P (x) = ∫ (1.85 + 0.12x - 0.0024x²) dx

= ∫1.85 dx+∫0.12x dx+∫ (-0.0024x²) dx + C

= 1.85x + 0.06x² - 0.0008x³ + C

Initial condition at P (0) = - 65

where x (0), P (x) = - 65

we have:

-65 = 1.85 (0) + 0.06 (0) ² - 0.0008x (0) ³ + C

-65 = 0 + 0 - 0 + C

-65 = C

C = - 65

P (x) = 1.85x + 0.06x² - 0.0008x³ - 65

Question:

A small tie shop finds that at a sales level of x ties per day its marginal profit is MP (x) dollars per tie, where

MP (x) = 1.85 + 0.12x - 0.0024x². Also, the shop will lose $65 per day at a sales level of x = 0. Find the profit from operating the shop at a sales level of x ties per day.

Answer:

P (x) = 1.85x + 0.06x^2 - 0.0008x^3 - 65

Step-by-step explanation:

MP (x) = 1.85 + 0.12 (x) - 0.0024 (x^2)

At sales level, x=0, loss = $65

P (0) = - $65

Integrating MP (x) with respect to x

P (x) = ∫MP (x) dx = ∫ 1.85 + 0.12 (x) - 0.0024 (x^2)

P (x) = 1.85x + 0.06x^2 - 0.0008x^3 + C

Therefore:

P (x) = 1.85x + 0.06x^2 - 0.0008x^3 - 65