The oil tank problem

A fuel-oil tank is 10 feet long and has flat ends that are perpendicular to the ground surface. Cross-sections parallel to the flat ends have the shape of the ellipse x2/9 + y2/36 = 1. If the fuel oil in the tank is 9 feet deep, what is the volume of the fuel oil in the tank?

Show the integrals you use to solve this problem. You MAY do a numerical integration, rather than applying the Fundamental Theorem.

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The oil tank problem Solution

The problem boils down to finding the area of the ellipse from the "bottom" (at y = -6) up to a depth of 9 feet (at y = 3), and then multiplying by the length, 10. Solving the equation for x gives x = (1/2)(sqrt(36 - y2)).

The integral of this function from -6 to 3 gives us half of the area of the ellipse (from the vertical axis to the right hand part of the curve).

We then double this value to get the entire area of the region in question, and then multiply by 10 to get the volume of the tank.

Numerical calculations give us an answer of 454.9 cubic feet.

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