Question

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?

Answer 1

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