Question

Profit Suppose that the daily profit (in dollars) from the production and sale of x units of a product is given byP180xx210002000At what rate per day is the profit changing when the number of units produced and sold is 100 and is increasing at a rate of 10 units per day

Answer 1

Answer:

The answer is "1798".

Explanation:

$\to p=180x-\frac{x^{2}}{1000}-2000$

In order to find the rate of profit increase each day, we differentiate between the money demand function and the time t.

$\to \frac{dp}{dt}=180\frac{dx}{dt}-\frac{2x}{1000}\frac{dx}{dt} \\\\\to \frac{dp}{dt}=\frac{dx}{dt}\left (180-\frac{2x}{1000} \right ).................(1)$

Calculate $\frac{dp}{dt}$ when $x=100$

$\frac{dx}{dt}=10$  (Extension rate of produced and delivered units per day)

$x=100 \ and\ \frac{dx}{dt}=10 ......... in \ \ eq(1)\\\\\frac{dp}{dt} = 10\left (180-\frac{2(100)}{1000} \right )$  

    $=10\left (180-0.2\right ) \\\\=1798$