Dear Mr. Brent Trachte,

We are writing to you in response to your latest letter, and we admit, we did not expect to hear from you so soon. We are disappointed that all of our hard work on your last water tank was not needed but we still are willing to help you out with this problem of your new water tank's calibration, expecting of course more money from your company. If you are curious as to why Rachel, Bryan and I are all writing to you this time, it is because Rachel hired Bryan and I to work for her since we proved to be wonderful Turkey tank problem solvers. Since Bryan was an illegal alien, and both Rachel and I wanted to help him out, we all became Mormons and we both married him. But that is beside the point of this letter.

In your letter you stated that you have all the measurements and the volume of your new cylindrical tank. The only problem was that you could not properly control how much water was leaving the tank at a time. Our company solved this problem for you in two ways. We first kept the hash marks the same as you gave them to us and recalculated the number of gallons at each mark, and secondly, we kept the numbers the same and recalibrated where they go on your tank's gauge. You can thank us again for our hard work and we would appreciate separate checks mailed to the address above.

Now for the details, we assumed that the 10 hash marks on the tank are each evenly spaced 1 foot apart from each other, and the top hash is at the top of the tank. (See Table 1). We also used the conversion that 1 gallon equals 0.1337 ft³. We labeled each hash mark from the bottom of the tank to the top with the numbers 1 through 10. (See Figure 1). After we labeled the diagram of your tank we first started to solve for the gallons at each mark when we keep your hash marks the same.

Hash Mark	Ft (h) from bottom	Hash Mark	Ft (h) from bottom
1	1	6	6
2	2	7	7
3	3	8	8
4	4	9	9
5	5	10	10

Figure 1. Labeled hash marks of water tank.

We then broke your water tank into different sections, using a triangle and the height (h) of the hash marks to find the new volumes at those marks. (See Figure 2), where h is the height of the hash mark from the bottom of the tank, 2J is the angle of the triangle and 5 is the radius of the cross section.

Figure 2. Cross section of water tank.

To find the volume at each hash mark, we had to combine 2 simple formulas that are found in the Hughes-Hallett Gleason 2nd Edition Single Variable Calculus book. These formulas are for Area of a triangle, which is

where b is the base of the triangle, in this case the square root of (10h-h²), and h is the height of the hash mark, and the area of a "pie slice" or a sector of a circle, which is

where r is the radius, in this case 5 and J is the angle of the pie slice [*comment on usage of variables]. The area of the pie slice minus the area of the triangle in it equals the area under the hash mark. We plug in our dimensions and now have the formula

Now, to find the volume at a given height, we must multiply the area of the section and the length of the tank together, plugging in for h the height of the hash mark we are working with. Now we have the equation

where V is the volume of the section and h is the height of the section. We solved for the volumes of the bottom half of the tank at hash marks 1 through 5 and then reflected those volumes to solve for the top half hash mark volumes. We did this using the formula Rachel derived, which is

where V_n is the volume we are solving for at the hash mark n, and 5 < n < 10. The results were in cubic feet so we converted to gallons using,

We now have completed the first part of your problem and found the number of gallons at each hash mark if we keep the hash marks the same. (See Table 2).

Hash Mark/height (h)	Number of gallons from bottom	Number of ft3 from bottom
1	489.257	65.4
2	1338.5	178.92
3	2372	317.07
4	3511.51	469.39
5	4700.46	628.32
6	5889.42	787.25
7	7028.92	939.57
8	8062.42	1077.72
9	8911.67	1191.24
10	9400.92	1256.64

We now have to work out the second way of solving your problem by keeping the numbers the same at each hash mark, but simply change where the marks go on the gauge. First we converted the gallons at each hash mark into ft³ , using the same conversion as before, to find the new hash marks in feet which we are sure you prefer. (See Table 3).

Hash Mark	Gallons	Ft3
1	918.2	122.73
2	1836.4	245.47
3	2754.6	368.2
4	3768.2	490.93
5	4591	613.66
6	5509.2	736.39
7	6427.4	859.12
8	7345.6	981.85
9	8263.8	1104.58
10	9182	1227.31

We then plugged our volume formula

into the graphing calculator as a function of h and graphed it. We then plugged in our h values. The points where our h values intersected the volume graph are the heights (h) where the new hash marks would be placed on your water tank. A fellow calculus classmate named Kevin helped us to figure this out. The heights of the new hash marks are listed in Table 4.

Hash Mark	Height in ft from bottom, (h)
1	1.53944
2	2.49846
3	3.34337
4	4.13695
5	4.90832
6	5.67746
7	6.46363
8	7.29259
9	8.21424
10	9.41917

Using our new hash marks, it seemed as though the tank is actually bigger than you thought. You stated that the volume of the water tank was 9,182 gallons. Using the formula

where L is the length of the tank, r is the radius and V is the volume, we found that the actual volume is 9,401.3 gallons. No matter how hard you try, it seems like you always make mistakes. It seems like that was also the case with Rachel's boyfriend Jeremy, who also tried to help us but led us astray. But Jeremy is a lot better than Bryan. You see, Rachel had to get a boyfriend because Bryan is always late to everything. We are threatening to fire Bryan before our next project if he does not start to arrive on time.

We would again like to thank you for this opportunity to utilize our extraordinary mathematical abilities.

Your always helpful, but awaiting their payment, problem solvers,

Rachel Wilson

Juliet D'Angelo

Bryan Ashby

P.S. We included a picture of us (with Mickey in place of Bryan, since he's always late), since you are always generous enough to send pictures of your water tanks.