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Problem 48 Hard Difficulty

Changing dimensions in a box The lengths $a, b,$ and $c$ of the
edges of a rectangular box are changing with time. At the instant
in question, $a=1 \mathrm{m}, b=2 \mathrm{m}, c=3 \mathrm{m}, d a / d t=d b / d t=$
$1 \mathrm{m} / \mathrm{sec},$ and $d c / d t=-3 \mathrm{m} / \mathrm{sec} .$ At what rates are the box's volume $V$ and surface area $S$ changing at that instant? Are the box's
interior diagonals increasing in length or decreasing?

Answer

$V_{t}=3 \frac{m^{3}}{s}, S_{t}=0 \frac{m^{2}}{s}, D_{t}=\frac{-6}{\sqrt{14}} \frac{m}{s}(\text {decreasing})$

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Video Transcript

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University of Wisconsin - Madison
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