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An environmental equipment supplier sells hemispherical holding ponds for treatment of chemical waste. The volume of a pond is V₁=1/2(4/3 π r₁³) , where r₁ is the radius in feet. The supplier 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 V₂=12 π r₂², where r₂ is the radius of the tank.
a. Since V₁=V₂ , write an equation that shows r₁ as a function of r₂ . Write an equation that shows r₂ as a function of r₁.

Respuesta :

varshamittal029

The equation for r₁ as a function of r₂ is r₁ = ∛ 18 ₂² and the equation for r₂ as a function of r₁ is r₂ = r₁ / 3 √r₁ / 2.

We are given that:

To fill a pond whose volume is given as V₁ = 1 / 2 (4 / 3 π r₁³), we need a cylindrical tank whose volume is given by V₂ = 12 π r₂².

Also, V₁=V₂

So, we get that:

1 / 2 (4 / 3 π r₁³) = 12 π r₂².

4 / 3 π r₁³ = 24 π r₂²

r₁³ = 18 r₂²

r₂ = √ ( r₁³ / 18)

r₂ = r₁ / 3 √r₁ / 2

Now, r₁³ = 18 r₂²

r₁ = ∛ 18 ₂²

Therefore, the equation for r₁ as a function of r₂ is r₁ = ∛ 18 ₂² and the equation for r₂ as a function of r₁ is r₂ = r₁ / 3 √r₁ / 2.

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