Aunt and Uncle's fuel oil tank dip stick problem?

Question

This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I am looking for is other WAYS to obtain the solution. I've been told there exists a solution using only arithmetic, but have never figured it out. Other solutions using ordinary calculus, trigonometry, algebra of conic sections, and so on are also possible. The problem is usually stated in the form of a letter from an Aunt and Uncle: > Dear niece/nephew, How are things > going for you and your folks? We hear > you are doing quite well it school. > Keep it up! Given this success, we > were hoping you could help us figure > out a little dilemma. As you know, > our home is heated by fuel oil, and we > have a big tank buried in the side > yard. The tank is a cylinder, 20 feet > long and 10 feet in diameter, lying on > its side five feet deep, with a narrow > tube coming to a fill cap at ground > level. Your uncle has a 15 foot > length of old pipe that we'd like to > utilize as a dip stick in order to > know when we are getting close to > needing a fill-up. We know that 0 > feet is empty, 5 feet is half full, > and 10 feet is completely full. > Trouble is, we don't know how to mark > any other points. We are pretty sure > they will not be uniformly spaced. > What we really want is to know, within > the nearest 0.01 foot, where to mark > the dip stick for every multiple of > 10% from 0% to 100%. Can you figure > this out for us? Of course, we will > want to see details of your solution > and check it ourselves, and it would > especially help if you could draw us a > scale model of the dip stick. Love, > Auntie Flo and Uncle Jim That last sentence shows the teacher influence on the problem. So, my challenge to this community is not to find any old solution, but to find the solution at the lowest possible grade level, so to speak. Thanks. UPDATE: To those who are focusing in on the .01 feet accuracy, I apologize. The intent was merely to state, it is acceptable to estimate. If the exact answer is sqrt(2)*pi/2 or some other silly thing, go ahead and just write 2.22 feet, for example.

Answer 1

You are out of your mind. This problem amounts to finding the area of a circle say x^2 + y^2 = R^2 between y=-R and arbitrary y. You're going to need calculus for that, there just isn't any other option. Prove me wrong.

Answer 2

Answer 3

Answer 4

the solution is not as complicated as the presentation!

Well let's look at this logically.

Forget the cylinder. We can work with just a slice/wafer. The ratios of areas on the slice are identical to the ratios of the volume.

Draw the slice - a circle with a vertical diameter and a horizontal diameter.

Now erase leave only the quarter in the upper right.

The point at the right angle is (0, 0), the point above, intersection of radius and circle, is (0, 5), the point at other intersection of radius and circle is (5, 0).

The area of the quarter is 6.25π.

The formula of the curve is y = √(25 - x²)

To determine the 10% mark∫

∫^x₀ (that's the integral from 0 to x) √(25 - x²) dx = 0.0625π (that's 10% of the area)

For the 20% mark do the same calculation for (0.20)(0.625π)

Continue with 30% and 40%.

Once you have the values you can do the same distances in the opposite direction.

OK? Ask if not.