# At a Glance - How to Solve a Math Problem

There are three steps to solving a math problem.

1. Figure out what the problem is asking.

2. Solve the problem.

Apply these steps to a sample optimization problem .

### Sample Problem

A 12-foot rope is to be cut into two pieces. One piece will lay across the bottom of a wall hanging, and the other piece will lay along the side of the wall hanging.
Where should the rope be cut to maximize the area of the wall hanging?

1. Figure out what the problem is asking.

The problem starts out with a 12-foot rope, which is to be cut into two pieces:

The pieces will then be rearranged like this to frame the wall hanging:

The problem says "where should the rope be cut to maximize the area of the wall hanging?" There are two things in here we need to translate into math.

The first thing is the phrase "where should the rope be cut." We need a mathematical way to represent this. Use a variable x that measures how many feet from the left end we cut the rope:

The second thing is the phrase "the area of the wall hanging". If we cut the rope x feet from the left end, the remaining piece of rope will be 12 – x feet long.

The wall hanging will look like this:

It's area will be

A(x) = x(12 &ndash; x) square feet

We've finally figured out what the problem is asking: find the value of x at which A attains its maximum value.

2. Solve the problem.

Now that we've translated the problem into math, this part isn't too bad. We have a function

A(x) = x(12 &ndash; x) = 12x – x2.

We need to find where A has a maximum. We start by taking the derivative:

A'(x) = 12 &ndash; 2x.

This is zero when x = 6, so that's our only critical point.

Use the second derivative test to see if this critical point is a maximum or a minimum. The second derivative is

A"(x) = -2

which is less than zero, so the function A is concave down. This means A must attain its maximum at the critical point x = 6.

Translating back into English, we want to cut the rope 6 feet from the end, or exactly in half.

P.S. A word about endpoints: In this problem, we know 0 ≤ x ≤ 12, because we can't cut the rope at a point that's not on the rope in the first place.

We can't have x = 0 or x = 12, however. If we did, there won't be any wall hanging because one of its dimensions would be zero.