Question

I'm completely lost about to work these problems

Answer 1

2) The formula for the area of a circle is: A = $\pi r x^{2}$
Differentiate both sides to get: $\frac{dA}{dt} = 2 \pi r \frac{dr}{dt}$
where dA/dt is the rate of change of the area of the circle with respect to time, and dr/dt is the rate of change of the radius of the circle with respect to time.
We want to know dA/dt when r = 3 and dr/dt = 2, so plug those into the equation to get: $\frac{dA}{dt} =2 \pi (3)(2) = 12 \pi feet per second.$

3) The amount of revenue gained in a week is equal to the price per good times the number of goods sold, or R = x*p. Thus, R = x(150-.002x) = 150x - .002x^2. Differentiate both sides to get: $\frac{dR}{dt} = 150 \frac{dx}{dt} - .004x \frac{dx}{dt}$, where dR/dt is the rate of change of revenue and dx/dt is the rate of change of production.
We want dR/dt when dx/dt = 200 and x = 2000, so plug those in to get: $\frac{dR}{dt} = 150(200) - .004(2000)(200) = 28400$