In this set we want to consider the problem
context of the storage of fluids in cylindrical tanks that have
been installed laying on their side. The common problem characteristic
is that the **DEPTH **of the fluid is known, determine the
volume.

**First Situation**

This problem was seriously proposed to one of my EMT 725 students by their Superintendent. At the Superintendent's home the furnace used fuel oil stored in an underground tank. It was known that the tank was installed level on its side and that it was 36 inches in diameter and 48 inches long. Using a stick dipped through the fill tube, the superintendent determined he had 10 inches of oil in the tank. He really did NOT want to know how to calculate the amount of oil. He knew from experience that it was February and he would need about 40 gallons of oil to finish the season. Would he have enough oil?

An **estimate **may suffice. For example,
the surface of the oil is a rectangle 48 incles long and a little
less than 18 incles wide. So the volume would be **less than
**48 X 36 X 10 cubic inches. This is the volume of rectangular
parallelepiped. At 231 cubic incles per gallon, this given an
overestimate of 75 gallons. In fact if we approximate the oil
in the tank by a trianglular prism with altitude of 10 and base
of 36 and lenght of 48 its volume is half of this parallelepiped,
or 37.5 gallons estimated. That estimate is probably pretty close,
but who would want to assure their boss that he has enough oil
in the tank to finish out the winter on the basis of this estimate?

We may want to **calculate** the volume.

How many gallons of oil are in the gas tank?

The volume of a cylinder is the area of its
base times the height. Since the tank is lying on its side, the
base is the part of the circle where the 10 inches of oil are
located. The drawing below is a diagram of the base.

The area of the base is the sector of the circle minus the area of the triangle. Since the radius of the circle is 18 and the height of the oil is 10, h=8. X can be found using the Pythagorean Theorem.

64+x^2=324

x^2=260

x=2(sqrt(65))

The area of the triangle is (1/2)bh. From the picture, we see that x= (1/2)b. The area of the triangle is xh.

= 2(sqrt(65))(8)

=16(sqrt(65)) in^2

To find the area of the sector, we must first find theta. Theta can be found using the arcos.

cos^-1 (theta) = 8/18

theta=1.11 radians

The area of the sector is (1/2) r^2 (theta).
From the picture, the angle marked theta is actually half of
the angle of the sector. Thus the area of the sector is r^2 (theta).

=(18^2)(1.11)

=359.72 in^2

The area of the oil is the area of the sector
minus the area of the triangle.

=359.72-16(sqrt(65))

=230.72 in^2

The volume of oil is the area of the oil times the length of the
tank.

=230.72 * 48

=11,074.67 in^3

However, the question asks for the number of gallons. The cubic inches must be converted to gallons. There are 7.481 gallons/cubic foot.

(11,074.67 in^3/1) * (1 ft^3/1,728 in^3) * (7.481 gallons/1 ft^3) = 47.94 gallons

There are approximately 48 gallons of oil in the tank.

**Extension**: This
problem was extremely lengthly. I would not want to do this every
time I wanted to know how much oil was in the tank. It would
be more time efficient to calibrate the stick used to measure
the depth of the oil so that the number of gallons could be read
off of it.

To make a calibrated stick, I created an Excel
spreadsheet that calculates the amount of oil in gallons for each
1/2 inch. The stick would be made by placing a mark at every
1/2 inch. The mark would be labeled with the number of gallons
for that particular 1/2 mark.

Click here to view
the spreadsheet.