When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.02 cm divided by min0.02 cm/min. at what rate is the plate's area increasing when the radius is 4040 cm?

Since the plate is circular, therefore the area of the plate is jut equal to the area of a circle, so:

Area of plate = πr² = A

Taking the derivative:

dA / dr = 2πr - - - > 1

By the idea of partial differentiation, the equation can also take in the form of:

dA/dt = dA/dr x dr/dt - - - > 2

Where we are given that:

change in radius over time = dr/dt = 0.02 cm/min

change in area with changing radius = dA/dr = 2πr - - - > from equation 1

at r = 40

dA/dr = 2π (40) = 80π

Substituting all the known values into equation 2:

dA/dt = (80π) (0.02)

dA/dt = 1.6π cm^2 / s = 5.03 cm^2/s