Answer:
The rate of change in surface area when r = 20 cm is 20,106.19 cm²/min.
Step-by-step explanation:
The area of a sphere is given by the following formula:
[tex]A = 4\pi r^{2}[/tex]
In which A is the area, measured in cm², and r is the radius, measured in cm.
Assume that the radius r of a sphere is expanding at a rate of 40 cm/min.
This means that [tex]\frac{dr}{dt} = 40[/tex]
Determine the rate of change in surface area when r = 20 cm.
This is [tex]\frac{dA}{dt}[/tex] when [tex]r = 20[/tex]. So
[tex]A = 4\pi r^{2}[/tex]
Applying implicit differentiation.
We have two variables, A and r, so:
[tex]\frac{dA}{dt} = 8r\pi \frac{dr}{dt}[/tex]
[tex]\frac{dA}{dt} = 8*20\pi*40[/tex]
[tex]\frac{dA}{dt} = 20106.19[/tex]
The rate of change in surface area when r = 20 cm is 20,106.19 cm²/min.
