rate of change in cost, revenue, and profit is needed

Question

Suppose that for a company manufacturing calculators, the cost, and revenue equations are given by

C = 80000 + 20x, R = 300 - x²/30,

where the production output in one week is x calculators. If the production rate is increasing at a rate of 500 calculators, dx/dt = 500, when the production output is 6000 calculators, that is x = 6000. 

Find rate of change in cost dC/dt =

Rate of change in revenue dR/dt =

Rate of change in profit dP/dt =

Answer 1

1. Find rate of change in cost (dc/dt) :

We will take the cost function and find its derivative with respect to time: dc/dt = 20*(dx/dt)

Plugging in the value for dx/dt and we get: dc/dt = 20*(500) = 10,000.

2. Rate of change in revenue dr/dt:

We will take the revenue function and find its derivative with respect to time: dr/dt = - (2x/30)*(dx/dt) ->

dr/dt = - (x/15)*(dx/dt).

Plugging in the value for dx/dt and x, we get: dr/dt = - (6000/15)*(500) = - 200,000.

3. To find dp/dt, we need to find the profit equation. That is found by using the following relationship: P = R - C

P = 300 - (x²/30) - (80,000+20x) -> P = -(x²/30) - 20x - 79,700. Now we can take the derivative with respect to time: dp/dt = - (x/15)(dx/dt) - 20(dx/dt).

Plugging in the known values and we get:

dp/dt =-200,000 - 10,000 -> dp/dt = - 210,000