2022-07-14

Finding the volume of the pyramid and also determining the increase and decrease of its volume
The volume of a pyramid with a square base x units on a side and a height of h is $V=\frac{1}{3}{x}^{2}h$.
1) Suppose $x={e}^{t}$ and $h={e}^{-2t}$. Use the chain rule to find ${V}^{1}\left(t\right)$.
2) Does the volume of the pyramid increase or decrease as t increases.

furniranizq

Expert

Step 1
$V=\frac{1}{3}{e}^{2t}{e}^{-2t}=\frac{1}{3}{e}^{2t-2t}=\frac{1}{3}{e}^{0}=\frac{1}{3}$
Thus ${V}^{\prime }\left(t\right)=0$. But if you insist on using the chain rule then, we apply the product rule and then the chain rule
Step 2
${V}^{\prime }=-\frac{2}{3}{e}^{2t}{e}^{-2t}+\frac{2}{3}{e}^{2t}{e}^{-2t}=0$
Clearly the volume of the pyramid clearly is a constant so doesn't increase or decrease.

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