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.
The Situation
This problem was seriously proposed to one of my students in this course 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?
Estimate
An estimate may suffice. For example, the surface of the oil is a rectangle 48 incles long and a little less than 36 incles wide. Looking at the end of the cylinder, we could estimate the volume in cubic inches by estimating the area of some cross-section and multiplying by 48. The length of a chord at a distance of 8 inches from the center of a circle of radius 18 inches is
inches or approximately 32 inches (Use the Pythagorean relationship).
So the volume would be less than 48 X 32 X 10 cubic inches. This is the volume of rectangular parallelepiped. At 231 cubic incles per gallon, this gives an overestimate of 66.5 gallons. In fact if we approximate the oil in the tank by a trianglular prism with altitude of 10 and base of 32 and lenght of 48 its volume is half of this parallelepiped, or 33.25 gallons estimated. That estimate is an underestimate. Since we have one overestimate and one underestimate, each closely fit to the desired shape, perhaps a refined estimated would be the midpoint or the arithmetic mean. That is approximately 50 gallons.
Can you propose other estimation strategies?
Measure
The superintendent has, in fact, measured the oil in the tank. He needs, however, a different metric. He needs to know "How many gallons?" for his measure rather than the depth in inches.
If we had a measure stick for this tank marked in gallons, we could measure directly. Of course, not being at the tank and not having a calibrated measure stick limits this strategy.
New Problem: How would one create such a measure stick?
A measure stick calibrated in gallons could have been created at the time the tank was filled from empty to full. Simply, put in 10 gallons at a time and mark off the depth on the stick for each 10 gallons.
Alternatively, software such as GSP could model the situation and calculate values.
Or, we could derive a function and use it to calibrate the measure stick.
Function for the Volume
A more accurate procedure is to develop a function that allows conversion from inches of depth to gallons of volume.
Derive a function f(x) where x is the measure of the depth in inches and f(x) is the volume in gallons. With f(x) we can generate a graph for 0 < x < 36. If we had such a function it could be implemented with a spread sheet or with a graphing program or tool. Further, it should be possible to include parameters for the radius and length of the tank in order to solve problems other the one given here.
HINT:
The segment of a circle is the region bounded by a chord and its minor arc. The sector of the circle is bounded by the minor arc and the two radii.
Area of segment = Area of sector - Area of triangle
We find an expression for the area of the sector in terms of the depth and multiply by 48 to get the volume in cubic inches. Then we devide by 231 to convert cubic inches to gallons.
[Note: The picture seems to imply that 0 < x < 18 but it will work for 0 < x < 36.]
If you want to use it, a GSP file is available with this set-up. Page 2 of the GSP file has a solution to allow you to compare and verify your derivation.
Once we have a function f(x) for the volume, a graph such as this can be shown.
Likewise, the function could be used to generate a spreadsheet table.
Click Here for the Graphing Calculator file that generated this graph.