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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 \(r_{1}\)=\(r_{2}\)?

Verified Answer :

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

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

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