A wire 370 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths (in inches) of the two pieces of wire (to the nearest tenth of an inch)?

To solve the problem, we need to find the lengths of the two pieces of wire. Let's break it down into steps:

### Step 1: Set up the problem
Let:
- \( x \) be the length of the piece of wire used to form the square.
- \( y \) be the length of the piece of wire used to form the circle.

The total length of the wire is 370 inches, so:
\[
x + y = 370
\]
The area of the square and the area of the circle must be equal.

### Step 2: Express the areas
1. **Square**: The perimeter of the square is \( x \). Since the perimeter of a square is \( 4s \) where \( s \) is the side length, we have:
   \[
   s = \frac{x}{4}
   \]
   The area of the square \( A_{\text{square}} \) is:
   \[
   A_{\text{square}} = s^2 = \left(\frac{x}{4}\right)^2 = \frac{x^2}{16}
   \]

2. **Circle**: The circumference of the circle is \( y \). The relationship between the circumference and the radius \( r \) is given by \( y = 2\pi r \). Solving for \( r \), we get:
   \[
   r = \frac{y}{2\pi}
   \]
   The area of the circle \( A_{\text{circle}} \) is:
   \[
   A_{\text{circle}} = \pi r^2 = \pi \left(\frac{y}{2\pi}\right)^2 = \frac{y^2}{4\pi}
   \]

### Step 3: Set the areas equal
Since the areas are equal:
\[
\frac{x^2}{16} = \frac{y^2}{4\pi}
\]

### Step 4: Solve for \( x \) and \( y \)
We can solve for \( y \) in terms of \( x \) using the equation \( x + y = 370 \) and substitute it into the area equation.

\[
\frac{x^2}{16} = \frac{(370 - x)^2}{4\pi}
\]

Multiply both sides by \( 16 \times 4\pi \) to eliminate the fractions:

\[
4\pi x^2 = 16(370 - x)^2
\]

Expand and solve for \( x \):

\[
4\pi x^2 = 16(136900 - 740x + x^2)
\]

\[
4\pi x^2 = 2190400 - 11840x + 16x^2
\]

This is a quadratic equation in \( x^2 \). After solving this equation, we can determine \( x \), and then find \( y = 370 - x \).

Let's solve this numerically.The lengths of the two pieces of wire are approximately:

- **Square piece**: \( 196.2 \) inches
- **Circle piece**: \( 173.8 \) inches

These lengths are rounded to the nearest tenth of an inch.

Would you like more details on the process, or do you have any questions?

Here are five related questions you might find interesting:
1. How would the solution change if the total wire length was different?
2. What would happen if the square and circle had different areas instead of the same?
3. Can you calculate the side length of the square and the radius of the circle based on these wire lengths?
4. What is the relationship between the perimeters and areas of these two shapes?
5. How can this problem be solved using a different approach, such as optimization?

**Tip:** For geometry problems, always try to express all variables in terms of one another using known geometric formulas, which simplifies the solving process.

Explore how to calculate the lengths of two wire pieces, one formed into a square and the other into a circle, both having equal areas. This problem involves geometry and quadratic equations, providing step-by-step solutions and numerical results. Suitable for advanced high school students.

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