Dear Brent,

I was delighted to receive another letter from you notifying me of your decision to grant me one more opportunity to make a contribution to the well-being of the turkeys on your farm. You asked me to figure out a way to cut an overflow pipe, so that it would fit snug with no holes or gaps when you attach it to the bottom of your water tank.

You also mentioned that you had 4 flat sheets of metal, each 1.6 feet long. You wanted to trim one end of each of the sheets and then join them together at the sides, so that you could make them into a pipe, which is 2 feet wide.

It is great that you had figured out that you could cut each of these 4 sheets the same. The problem you were facing was to find out a way to cut these sheets.

I am happy to tell you that I succeeded to solve the problem. The formula for the curve, through which you need to cut each sheet of metal, is

where U is the height of the pipe, starting point of which is the bottom of the water tank and which is measured in feet, and L is the length of the flat sheet of metal measured also in feet. The curve of this function is shown on Figure 1. So here is what you have to do. Cut each sheet of metal the way it is shown on the diagram below (Figure 1), and then put all 4 sheets together the way it is show on the second diagram (Figure 2).

Figure 1: A flat sheet of metal, 1.6 feet long (green curve shows how to cut the sheet).

 

Figure 2: Four flat sheets of metal put together.

In case you are wondering how I managed to come up with such a solution, I'll show you how amazing calculus is.

Let us look at your tank and pipe from the side.

Figure 3: The tank and the pipe(view from the side).

If we cut the tank and the pipe through the green line shown on Figure 3, we will get the following diagram.

Figure 4: The tank and the pipe cut through the center of the tank (view from the front).

As you can see from the diagram (Figure 4), the values of U will tell us how to cut the pipe. So all we have to do is to figure out a general formula for U.

There is a formula for a circle given in my calculus textbook (Calculus Single Variable, second edition, Hughes-Hallet and Gleason, et al.). It says that

where 5 stands for the radius of the tank measured in feet; (5 - U) represents the distance toward the bottom of the tank measured in feet; and y is the distance toward the side of the tank measured also in feet.

Because we'll need to use "y²" later on, we'll solve this equation for y². Thus,

Now we need to look at our water tank and the pipe from below.

Figure 5: Then tank and the pipe (view from below).

As you see on Figure 5, there is another circle, with radius 1 this time, measured in feet, that we can write a formula for.

The formula for this circle would be

where 1 stands for the radius of the circle; y, just as before, is the distance toward the side of the tank; and x is the distance toward the front of the tank.

Solving for y² again, we will get

Here we've got 2 formulas:

Since the left-hand sides of both formulas are equal to y2, we can say that the right-hand sides are equal.

Solving for (5 - U)², we will get

This means that

Solving for U, we will have

We have the general formula for the height of the pipe (U) in terms of the distance toward the front of the tank (x). But this formula still has the pipe rolled up, and you want to measure when the pipe is unrolled. Therefore, we need to figure out a general formula for U in terms of the length of the metal flat sheet, the formula mentioned in the beginning of my letter.

Now you see the only difference between that formula U = 5 - root(24 + sin²(L)) and this formula U = 5 - root(24 + x²) is that the former has sin²(L), and the latter has x². Thus, if we can show that

which also means that

we'll have the formula that we need, a formula for the height of the pipe (U) in terms of the length of the flat sheet of metal (L).

Let us look at the following diagram.

Figure 6: Diagram of the pipe (views when the pipe is rolled up and when it is unrolled).

As you can see from the diagram (Figure 6), when the pipe is rolled up, the length of the flat sheet (L) becomes the arclength (A), but the value of L remains the same. In other words,

A formula for the arclength is also given in our calculus textbook (Calculus Single Variable, second edition, Hughes-Hallet and Gleason, et al.). It says that

where A represents the arclength measured in feet.

I suppose that the right hand side of the formula looks quite unfamiliar to you. So, instead of wasting my time and paper on explaining unnecessary stuff like "integrals" and "derivatives", I'll just give you the solution for x, which is

Since A = L, we can say that

Thus,

Now, if we substitute x with sin(L) in the general formula for U, we will have

The only thing we have to do now is punch a couple of buttons on the calculator.

When L = 0 (which means you're on the one edge of the metal sheet),

When L = 1.6 (which means you're on the other edge of the metal sheet),

Now you see how I came up with the diagram on Figure 1.

In case if you feel like you need more values of U, so that the cutting would be better, please just punch the buttons on the calculator couple more times and find out the values of U at L = 0.8 foot and at L = 1 foot.

I hope you are satisfied with the answer that I came up with. As for the payment, please send the paycheck to my math professor, Dr. Crannell. Without her help, I simply couldn't have solved this problem. The knowledge obtained from working on the problem is enough payment for me.

Sincerely,

Calculus 2
Franklin and Marshall College