Annie plots an ellipse on a computer screen. It has horizontal radius a and vertical radius b.?

Question

She investigates the rate of change of the area of the ellipse as a increases at 7mm/s and b decreases at 11 mm/s.
1. Assuming that a and b change smoothly, write the known rate and the wanted rate as derivatives.
2. Annie recalls that the the area of an ellipse is A=(pie- i cant type the symbol)times ab. use this relationship among the variables to get an equation for the wanted rate.
3. Annie sets the animation in motion. when a = 30 and b = 50 will the area be increasing or decreasing? at what rate?
4. Annie resets the animation and starts it in motion again. When a =40 and b=20, will the area be ingreasing or decreasing? At what rate?

Answer 1

I guess 1) just wants you to write: Want dA/dt, have da/dt and db/dt.
(From now on I'll use A' to mean dA/dt etc).
2)
A=π*a*b, so using the product rule, A' = πa'b + πab'. So A' = 7πb - 11πa.
3) Just substitute in: A' = 7*π*50 - 11*π*30 = 20π, so thats the rate it is increasing.
4) Again substitute in, A' = 7*π*20 - 11*π*40 = -300π, so now its decreasing at that rate.