Answer:
The answer is [tex]\frac{3}{10\pi } cm/min[/tex]
Step-by-step explanation:
Assuming the snowball is a perfect sphere, then if A denotes the surface area and D the diameter then:
[tex]A=4\pi r^{2} = 4\pi (\frac{D}{2} )^{2} =\pi D^{2}[/tex]
Differentiating wrt r we have:
[tex]\frac{dA}{dD} =2\pi D[/tex]
We are told that [tex]\frac{dA}{dt}= -6[/tex] and we want to find [tex]\frac{dD}{dt}[/tex]
By the chain rule we have:
[tex]\frac{dA}{dD}=\frac{dA}{dt}.\frac{dt}{dD}=\frac{\frac{dA}{dt} }{\frac{dD}{dt} }[/tex]
∴[tex]2\pi D=-\frac{6}{\frac{dD}{dt} }[/tex]
∴[tex]\frac{dD}{dt}=-\frac{6}{2\pi D}[/tex]
When D=10 then
[tex]\frac{dD}{dt} =-\frac{6}{10*2\pi } =-\frac{3}{10\pi }[/tex]
The sign (-) shows that the D is decreasing.
