I responded to Tom & Ray's request for help by putting up the following web page, and sending them a summary and a link. They printed several peoples' solutions, but (alas) not mine. I reproduce it below for your comfort and convenience:
Dear Tom & Ray,
The key to making this problem simpler is to measure the height of the dipstick from the CENTER of the tank like so:
So now we compute this integral: 
And we get this answer: 
So now we know the area of the circle as a function of h. We can set f(h) = 3/4(the area of the circle) to find out that the tank is 3/4 full when h is approximately 4.0397. In other words, the tank is 3/4 full when the dowel rod measures approximately 14 inches.
We can set f(h) = 1/4(the area of the circle) to find out that the tank is 1/4 full when h is approximately -4.0397. In other words, the tank is 1/4 full when the dowel rod measures approximately 6 inches.
I think it would be more useful to mark the dowel rod off in tenths. And then, under each marking, write down what it means about the the tank. Like so:
| Inches from bottom of rod | Percent full |
|
0
|
0 %
|
|
1
|
2 %
|
|
2
|
5 %
|
|
3
|
9 %
|
|
4
|
14 %
|
|
5
|
20 %
|
|
6
|
25 %
|
|
7
|
31 %
|
|
8
|
37 %
|
|
9
|
44 %
|
|
10
|
50 %
|
|
11
|
56 %
|
|
12
|
63 %
|
|
13
|
69 %
|
|
14
|
75 %
|
|
15
|
80 %
|
|
16
|
86 %
|
|
17
|
91 %
|
|
18
|
95 %
|
|
19
|
98 %
|
|
20
|
100 %
|
Key numbers to note: 5 inches = 1/5 full, 6 inches = 1/4 full, 10 inches = 1/2 full, 14 inches = 3/4 full, 15 inches = 4/5 full
INTERESTING FACT:
Except when the tank is nearly empty or nearly full, the relationship between the distance from the middle of the dowel and the amount of fuel is nearly linear! Observe:
Drive safely!
Click here to see the
original problem
Click here to go to Doug's
AP page
Click here to go to Doug's homepage