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Question: An environmental provider sells hemispherical hold...
An environmental provider sells hemispherical holding ponds for treatment of chemical waste. The volume of a pond is
`V_(1)`=1/2( 4/3pi x
`r_(1)`^3), where
`r_(1)` is the radius in feet. The provider also sells cylindrical collecting tanks. A collecting tank fills completely and then drains completely to fill the empty pond. The volume of the tank is
`V_(2)`=12 pie x
`r_(2)`^2, where
`r_(2)` is the radius of the tank.

Given that the volumes are equal,
`V_(1)`=
`V_(2)`, Write an equation that shows
`r_(1)` as a function of
`r_(2)`.

Is it
`r_(1)`=
`r_(2)`?

Answer 1

`\bf \begin{cases} V_1=(1)/(2)\left( (4)/(3)\pi\cdot x\cdot r_1^3 \right) \\\\ V_1=12\cdot \pi \cdot x\cdot r_2^2\\ --------------\\ V_1=V_2 \\\\ (1)/(2)\left( (4)/(3)\pi\cdot x\cdot r_1^3 \right)=12\cdot \pi \cdot x\cdot r_2^2 \end{cases} \\\\\\ \cfrac{1}{2}\left( (4)/(3)\pi\cdot x\cdot r_1^3 \right)=12\cdot \pi \cdot x\cdot r_2^2\implies \cfrac{2\pi x}{3}r_1^3=12\pi xr_2^2`


`\bf r_1^3=\cfrac{3}{2\pi x}\cdot 12\pi x r_2^2\implies r_1^3=\cfrac{3}{1}\cdot 6r_2^2 \\\\\\ r_1^3=18r_2^2\impliedby \textit{now, taking }\sqrt[3]{\qquad }\textit{ to both sides} \\\\\\ \sqrt[3]{r_1^3}=\sqrt[3]{18r_2^2}\implies r_1=\sqrt[3]{18r_2^2}`