# Area, Volume, and Arc Length

### Topics

There are three steps to solving a math problem.

1) Figure out what the problem is asking.

2) Solve the problem.

3) Check the answer.

### Sample Problem

A conical water tower is 20 feet tall and 14 feet in diameter at the top. The tower starts completely full and then half the water drains out. To the nearest foot, what is the depth of the remaining water?

Answer.

1) Figure out what the problem is asking.

We're told there's a conical water tower and given its dimensions. We really need the radius, not the diameter. The tower starts out full of water, so the volume of water is equal to the volume of the tower. After half the water drains out, the volume of water remaining will equal half the volume of the tower.

The question is at what depth* H* the volume of water equals half the volume of the tower.

Take horizontal slices of the tower, using *h* for the distance from the bottom of the tower to the slice. Then the volume of water in the tower is

We want to find the value of *H* for which

2) Solve the problem.

We need to get *r* in terms of *h* so we can do the integrals. Slice the cone down the middle to see the similar triangles.

We have

so

The volume of the tower is

So half the volume of the tower is

We need to find H such that

When we evaluate the integral on the left-hand side of this equation, we get

We set this equal to the volume of the tower and solve for *H*.

Rounding to the nearest foot,

*H* ≅ 16.

The water left in the tower will be approximately 16 feet deep.

3) Check the answer.

To check the answer, use a calculator to evaluate the volume of the tower:

Then use the calculator to evaluate the volume of water when it's 16 feet deep:

Since 525 is approximately half the volume of the full tower (), our answer is reasonable.