Questions: Find the area of the shaded region inside the square and between the two circles. Use π ≈ 3.14. The area of the shaded region is approximately □ m^2 (Type an integer or decimal)

Find the area of the shaded region inside the square and between the two circles. Use π ≈ 3.14.

The area of the shaded region is approximately □ m^2
(Type an integer or decimal)

Transcript text: Find the area of the shaded region inside the square and between the two circles. Use $\pi \approx 3.14$. The area of the shaded region is approximately $\square$ $\mathrm{m}^{2}$ (Type an integer or decimal)

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the area of the shaded region inside the square and between the two circles. The circles have a radius of 34 meters each, and the value of π is approximately 3.14.

Step 2: Calculate the Area of the Square

The side length of the square is equal to the diameter of the two circles combined. Since each circle has a radius of 34 meters, the diameter of each circle is $2 \times 34 = 68$ meters. Therefore, the side length of the square is $68 + 68 = 136$ meters. \[ \text{Area of the square} = \text{side length}^2 = 136^2 = 18496 \, \text{m}^2 \]

Step 3: Calculate the Area of One Circle

The area of one circle is given by the formula $ \pi r^2 $, where $ r $ is the radius. \[ \text{Area of one circle} = \pi \times 34^2 = 3.14 \times 1156 = 3622.64 \, \text{m}^2 \]

Step 4: Calculate the Total Area of the Two Circles

Since there are two circles, we multiply the area of one circle by 2. \[ \text{Total area of the two circles} = 2 \times 3622.64 = 7245.28 \, \text{m}^2 \]

Step 5: Calculate the Area of the Shaded Region

The area of the shaded region is the area of the square minus the total area of the two circles. \[ \text{Area of the shaded region} = 18496 - 7245.28 = 11250.72 \, \text{m}^2 \]