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nirvana33 [79]
3 years ago
5

webassign If a snowball melts so that its surface area decreases at a rate of 6 cm2/min, find the rate at which the diameter dec

reases when the diameter is 10 cm.
Mathematics
1 answer:
drek231 [11]3 years ago
3 0

Answer:

The answer is \frac{3}{10\pi }  cm/min

Step-by-step explanation:

Assuming the snowball is a perfect sphere, then if A denotes the surface area and D the diameter then:

A=4\pi r^{2} = 4\pi (\frac{D}{2} )^{2} =\pi  D^{2}

Differentiating wrt r we have:

\frac{dA}{dD} =2\pi D

We are told that \frac{dA}{dt}= -6 and we want to find \frac{dD}{dt}

By the chain rule we have:

\frac{dA}{dD}=\frac{dA}{dt}.\frac{dt}{dD}=\frac{\frac{dA}{dt} }{\frac{dD}{dt} }

∴2\pi D=-\frac{6}{\frac{dD}{dt} }

∴\frac{dD}{dt}=-\frac{6}{2\pi D}

When D=10 then

\frac{dD}{dt} =-\frac{6}{10*2\pi } =-\frac{3}{10\pi }

The sign (-) shows that the D is decreasing.

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