This problem comes from Car Talk. An 18-wheeler truck driver has an unreliable fuel gauge. His fuel tank is a cylinder that lies horizontally and is 20 inches in diameter.
He wants to know when his tank will be 1/4 full. He checks his fuel by putting a dip-stick vertically in his tank.
Solve for x in the diagram.
Solution to the Problem:
The answer is 5.96 inches.We are looking for the value of x (the height of the fuel on the dip-stick) so that the area of the sector below the triangle is 1/4 of the area of the circle (see the diagram below).
Find the length of the base of the triangle:
Thus, the area of the triangle is:
Now find the area of the sector:
So, the area of the section containing the fuel equals the area of the sector minus the area of the triangle:
Since we want this area to be 1/4 of the area of the circle:
Since we know the radius is 10 inches, the equation we want to solve is:
I substituted different values of x in order to narrow down the answer.
Since 25 pi = 78.539816, I was looking for the value of x that came closest to that number.
| when x = 5 | A = 61.4184 | |
| when x = 5.9 | A = 77.4387 | |
| when x = 5.95 | A = 78.3519 | |
| when x = 5.96 | A = 78.534831 | |
| when x = 6 | A = 79.2673425 | |
| when x = 7 | A = 97.99219 |
Click here for a calculus solution
Correctly solved by:
| 1. Tom Laidlaw | Vancouver, Washington |
| 2. James Alarie | Flint, Michigan |
| 3. Aubrey Reynolds |
Mountain View High School, Mountain View, Wyoming |
| 4. Marivi Bungay-Domingo | Philippines |