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?
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
and this is about 32 inches.
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 given 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.
Are there other estimation strategies that might come closer to the actual amount of oil?
Perhaps we want to measure the number of gallons of oil.
If we had a measure stick for this tank, marked in gallons, we could measure. We measure directly. Of course, not being at the tank and not having a calabrated measure stick limits this strategy. New Problem: How would one create such a measure stick?
GSP has tools to allow direct measure of areas. So it is possible to set up construction of a segment of a circle and measure it. True to the GSP spirit, any information on how the area was calculated is hidden. At the same time the areas can be plotted in a grid as the size of the segment is varied.
This GSP construction has been provided by Derek Reeves. When the point of intersection of the chord is animated along the perpendicular, direct measures of the area of the yellow segments can be read. These readings can then be plotted on a grid.
Some ingenuity will be needed to adjust the unit to the cylinder problems.
As Derek notes, this GSP sketch can be used to develop a model of the situation.
Click HERE to see this GSP File
CALCULATION
There is still a need to derive a calculation of the volume and to generate a function f(x) to see a graph. 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.
One 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.
