Expanding Sphere

Calculus Level 3

A sphere is slowly expanding in size. If the radius of the sphere is increasing at a rate of 4 cm per second, then how fast is the volume (in cubic cm per second) increasing when the diameter is 80 cm (to the nearest whole number)?


The answer is 80425.

Deva Craig by Deva Craig , 22, USA

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1 solution

Chew-Seong Cheong
Oct 31, 2017

Volume of a sphere with radius r r is given by:

V = 4 3 π r 3 d V d t = 4 π r 2 d r d t Note that r = 40 cm, d r d t = 4 cm/s = 4 π ⋅ 4 0 2 ⋅ 4 ≈ 80425 cm 3 /s \begin{aligned} V & = \frac 43 \pi r^3 \\ \frac {dV}{dt} & = 4 \pi r^2 \frac {dr}{dt} & \small \color{#3D99F6} \text{Note that }r=40 \text{ cm, } \frac {dr}{dt} = 4 \text{ cm/s} \\ & = 4\pi \cdot 40^2 \cdot 4 \\ & \approx \boxed{80425} \text{ cm}^3\text{/s} \end{aligned}

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