Mathematics

Problem

The temperature at the surface of an inflating balloon is given by

T(V)=100\sqrt{V}

, where

T

is measured in degrees Celsius and

V

(in cubic meters) is the balloon’s volume. The balloon remains spherical so that

V=\dfrac{4}{3}\pi r^{3}

, where

r(t)

, in meters, is the radius at time

t

(in seconds). When

r=5\text{ m}

and the radius is increasing at

0.20\text{ m/s}

, which of the following is closest to

\dfrac{dT}{dt}

? a)

7.1\times10^{2}\;^{\circ}\!\text{C/s}

2.2\times10^{1}\;^{\circ}\!\text{C/s}

1.4\times10^{2}\;^{\circ}\!\text{C/s}

1.4\times10^{3}\;^{\circ}\!\text{C/s}

Intuitive explanation

You're chaining together three rates: how temperature depends on volume, how volume depends on radius, and how radius changes with time. Use the chain rule to connect them— $\frac{dT}{dt} = \frac{dT}{dV} \cdot \frac{dV}{dr} \cdot \frac{dr}{dt}$ —then plug in the given values at the specific moment. This is a classic related rates problem where you differentiate each relationship and multiply the derivatives together.

Intuitive explanation

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