# 12.8.2. Maximum and Minimum Values

A wire having a length of cm is divided into two portions. A square was constructed from one portion and a circle from the other portion. Find the length of each portion if the sum of the two areas of the two figures is to be minimum.

• A The length of square portion is cm, and the length of circle portion is cm.
• B The length of square portion is cm, and the length of circle portion is cm.
• C The length of square portion is cm, and the length of circle portion is cm.
• D The length of square portion is cm, and the length of circle portion is cm.

### Example

A wire having a length of cm is divided into two portions. A square was constructed from one portion and a circle from the other portion. Find the length of each portion if the sum of the two areas of the two figures is to be minimum.

### Solution

Let the lengths of the two portions of wire be denoted by and .

Let the circle be constructed from the portion whose length is . If the circumference of the circle is , then the radius can be found by rearranging the formula for the circumference of a circle.

Let the square be constructed from the portion whose length is . The side length of the square will therefore be .

Now find the area of each shape.

Circle:

Square:

The total area to be minimized is therefore the sum of these two expressions.

Now differentiate with respect to .

Now set the derivative equal to 0 and solve for .

Hence, the length of the circle portion is .

The length of the square portion is

Hence, the length of the square portion is .

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