Water in a Tank - Integration

**rawkstar** · April 5th 2010, 04:08 PM

a tank on a water tower is a sphere of a radius of 50 feet. determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. use root finding capabilities of a graphing utility after evaluating the definite integral

Please Help

**skeeter** · April 5th 2010, 04:26 PM

Originally Posted by rawkstar

a tank on a water tower is a sphere of a radius of 50 feet. determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. use root finding capabilities of a graphing utility after evaluating the definite integral

Please Help

consider the circle $x^2 + y^2 = R^2$ centered at the origin.

rotation of this circle about the y-axis yields the sphere.

volume of the sphere ...

$V = \pi \int_{-R}^{R} R^2 - y^2 \, dy = \frac{4\pi}{3} \cdot R^3$

let $h$ = variable water level in the tank from the bottom.

$\frac{V}{4} = \frac{\pi}{3} \cdot R^3 = \pi \int_{-R}^h R^2 - y^2 \, dy$

... solve for $h$ .

once you determine $h$ , it should just take a little thought (no calculus) to determine the level where the tank is 3/4 full.

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