Question

Suppose that the daily profit (in dollars) from the production and sale of x units of a product is given by

rho=170x- x²/600 -1900.
At 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 6 units per day?_________per day

Answer 1

The profit is changing at a rate of $1016 per day when 100 units are produced and sold and the number is increasing by 6 units per day.

Step-by-step explanation:

The problem requires us to determine the rate at which profit is changing with respect to the number of units produced and sold per day, which is a problem of finding the derivative of the profit function with respect to time.

We are given the profit function ρ = 170x - x²/600 - 1900 and need to find the rate of change of profit (ρ) when x = 100 units and is increasing at 6 units per day. We compute this by taking the derivative of the profit function with respect to x to find dρ/dx and then multiply it by the rate of change of x with respect to time, dx/dt.

1. First, find the derivative of the profit function with respect to x: dρ/dx = 170 - (2/600)×x.
2. Next, evaluate this derivative at x = 100: dρ/dx = 170 - (2/600)×100 = 170 - 1/3.
3. Finally, multiply by the rate of increase of x: dx/dt = 6 units/day. So, dρ/dt = (170 - 1/3)×6 = 1018 - 2 = 1016 per day.

The profit is changing at a rate of $1016 per day when 100 units are produced and sold and this number is increasing by 6 units per day.