1955 AMC 12 Problem 36

Question

A cylindrical oil tank, lying horizontally, has an interior length of

10

feet and an interior diameter of

6

feet. If the rectangular surface of the oil has an area of

40

square feet, the depth of the oil is:

\sqrt5

2\sqrt5

3-\sqrt5

3+\sqrt5

either

3-\sqrt5

or

3+\sqrt5

The oil surface is a rectangle of length

10

, so its width is

4

In the circular end, a chord of length

4

lies

\sqrt{3^2-2^2}

from the center

The rectangular surface has length

10,

so its chord width in the circular cross-section is

\frac{40}{10}=4.

Half the chord is

2,

and the radius is

3,

so the distance from the center to the chord is

\sqrt{3^2-2^2}=\sqrt5.

A chord of this length can lie either below or above the center. Measured from the bottom of the tank, the corresponding depths are

3-\sqrt5

and

3+\sqrt5.

Thus, the correct answer is E.

Problem 36 in other years