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