ok whoever answers this right will get brainliest is 0.032 the same as 32%? and is 32/1000 the same as 32%?

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whereswoodruff

23.01.2022

Mathematics

$\text{given: }\dfrac{da}{dt} = 4 \text{ cm}^2\text{/sec} \text{ when } a = 100\text{ cm}^2 \\ \text{unknown: }\dfrac{dv}{dt} \text{ when } a = 100\text{ cm}^2$ we have info about sa, but not radius. the volume formula uses radius. we can radius in terms of sa so we can use our information given about sa for volume. $\begin{aligned} a& = 4\pi r^2 \\ r^2 & = \frac{a}{4\pi} \\ r & = \sqrt{ \frac{a}{4\pi} } \\ r & = \left( \frac{a}{4\pi}\right)^{1/2}\end{aligned}$ so relating sa to volume: $\begin{aligned} v & = \frac{4}{3}\pi r^3\\ v & = \frac{4}{3}\pi \left( \frac{a}{4\pi} \right)^{3/2}\end{aligned}$ differentiating both sides (using the chain rule): $\begin{aligned} \frac{dv}{dt} & = \frac{4}{3}\pi \cdot \frac{3}{2} \left( \frac{a}{4\pi} \right)^{\frac{3}{2}- 1} \cdot \frac{1}{4\pi} \frac{da}{dt} \\ \frac{dv}{dt} & = \frac{2\pi}{4\pi} \cdot \left( \frac{a}{4\pi} \right)^{1/2} \cdot \frac{da}{dt} \\ \frac{dv}{dt} & = \frac{1}{2} \cdot \left( \frac{a}{4\pi} \right)^{1/2} \cdot \frac{da}{dt} \\ \end{aligned}$ substituting information: $\begin{aligned} \frac{dv}{dt} & = \frac{1}{2} \cdot \left( \frac{100}{4\pi} \right)^{1/2} \cdot 4 \\ & =2 \sqrt{\frac{25}{\pi} } \\ & =2 \cdot 5 \sqrt{\frac{1}{\pi} } \\ & = \frac{10}{\sqrt{\pi}}\text{ cm}^3\text{/sec} \end{aligned}$

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