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.
b. You want to double the radius of the pond. How will the radius of the tank change?

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varshamittal029 varshamittal029 · Answer 1 · 2022-09-16T18:27:48+02:00

The radius of the tank should increase √ 2 times.

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.

This means that the volume of the pond should be equal to the volume of the cylinder.

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, we will double the radius of the pond.

So, r₁ will become 2 r₁

Substituting it, we get that:

R₂ = 2r₁ / 3 √2r₁ / 2

R₂ = 2r₁ / 3 √r₁

So, we can see that:

R₂ / r₂ = 2r₁ / 3 √r₁ / (r₁ / 3 √r₁ / 2)

R₂ / r₂ = 2 / √2 = √ 2

Therefore, the radius of the tank should increase √ 2 times.

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