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 of the two pieces of wire (to the nearest tenth of an inch)?

We are tasked with dividing a 370-inch long wire into two pieces: one piece is formed into a square, and the other into a circle. The areas of the square and the circle must be equal. Let's solve this step by step.

### Step 1: Let \( x \) be the length of the wire used for the square.
Thus, the length of the wire used for the circle will be \( 370 - x \).

### Step 2: Express the perimeter of the square and circumference of the circle.
- The perimeter of the square is \( x \), so the side length of the square is \( \frac{x}{4} \).
- The circumference of the circle is \( 370 - x \), so the radius of the circle is \( \frac{370 - x}{2\pi} \).

### Step 3: Express the area of the square and the circle.
- The area \( A_s \) of the square is:
  \[
  A_s = \left( \frac{x}{4} \right)^2 = \frac{x^2}{16}
  \]
- The area \( A_c \) of the circle is:
  \[
  A_c = \pi \left( \frac{370 - x}{2\pi} \right)^2 = \frac{(370 - x)^2}{4\pi}
  \]

### Step 4: Set the areas equal to each other.
Since the areas of the square and the circle are equal:
\[
\frac{x^2}{16} = \frac{(370 - x)^2}{4\pi}
\]

### Step 5: Solve for \( x \).
Multiply both sides by \( 16\pi \) to eliminate the fractions:
\[
\pi x^2 = 4(370 - x)^2
\]

Expand and solve the equation. I'll calculate the solution now.The positive solution for \( x \), which represents the length of the wire used for the square, is approximately 196.2 inches. 

### Step 6: Calculate the length of the wire used for the circle.
The length of the wire used for the circle is:
\[
370 - 196.2 = 173.8 \text{ inches}.
\]

Thus, the lengths of the two pieces of wire are:
- **Square:** 196.2 inches
- **Circle:** 173.8 inches

Would you like further details on the solution or have any other questions? Here are five related questions:

1. How does changing the total length of the wire affect the solution?
2. What if the areas of the figures were not equal—how would you approach the problem?
3. How would you solve this problem if you were dealing with a rectangle instead of a square?
4. How can we derive the radius and side lengths from the respective wire lengths?
5. Can this method be extended to other shapes, like triangles or ellipses?

**Tip:** When working with geometry problems, it's often helpful to sketch the figures to visualize relationships between lengths and areas.

Solve a geometry problem where a 370-inch wire is cut into two pieces to form a square and a circle with equal areas. This detailed solution covers concepts of area, perimeter, and circumference and uses quadratic equations to find the wire lengths. Ideal for students in Grades 9-12.

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