Related Rates

Calculus Level 3

A spherical snowball melts so that the surface area decreases at a rate of 0.5 cm 2 /min 0.5\text{ cm}^{2}\text{/min} . Find the rate at which the radius is decreasing (with respect to time) when the radius is 4 cm 4\text{ cm} .

Give your answer to 3 significant figures.


The answer is 0.00497.

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Denis Nadarević by Denis Nadarević , 23, Canada

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1 solution

Denis Nadarević
Dec 23, 2016

This is a related rates problem, an application to implicit differentiation.

We start off with the formula for the surface area of a sphere, which is: S A = 4 π r 2 SA= 4πr^{2} . Now, we pull off information from the problem:

d ( S A ) d t = 0.5 \frac{d(SA)}{dt}=0.5 , d ( r ) d t = ? \frac{d(r)}{dt}=? , r = 4 r=4

Now, differentiate with respect to t and then substitute r with 4 : d ( S A ) d t = d ( 4 π r 2 ) d t = 8 π ( 4 ) [ d ( r ) d t ] = 0.5 \frac{d(SA)}{dt}=\frac{d(4πr^{2})}{dt}=8π(4)[\frac{d(r)}{dt}]=0.5 .

Isolate d ( r ) d t \frac{d(r)}{dt} , solve, and round to 3 decimal places.

d ( r ) d t = 0.5 32 π = 0.004973592 ≈ 0.005 \frac{d(r)}{dt}= \frac{0.5}{32π} = 0.004973592 ≈ 0.005

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Great solution!

Note: We accept decimal answers to a 2% margin of error, so it's better to give it to 3 significant figures instead. I've updated the answer to 0.00497.

Calvin Lin Staff - 4 years, 5 months ago

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