Find the area of the shaded region. Use 3.14 for pi.

Steps to Solve

1. Calculate the area of the rectangle

To find the area of the rectangle, use the formula:

$$ \text{Area}_{\text{rectangle}} = \text{length} \times \text{width} $$

Here, the length is $18 , \text{in}$ and the width is $12 , \text{in}$:

$$ \text{Area}_{\text{rectangle}} = 18 \times 12 = 216 , \text{in}^2 $$

2. Calculate the areas of the circles

Next, we need to calculate the area of each circle using the formula:

$$ \text{Area}_{\text{circle}} = \pi r^2 $$

Using $\pi \approx 3.14$:

• For the circle with radius $6 , \text{in}$:

$$ \text{Area}_{\text{circle}_1} = 3.14 \times (6^2) = 3.14 \times 36 = 113.04 , \text{in}^2 $$

• For the circle with radius $9 , \text{in}$:

$$ \text{Area}_{\text{circle}_2} = 3.14 \times (9^2) = 3.14 \times 81 = 254.34 , \text{in}^2 $$

• For the circle with radius $12 , \text{in}$:

$$ \text{Area}_{\text{circle}_3} = 3.14 \times (12^2) = 3.14 \times 144 = 452.16 , \text{in}^2 $$

3. Sum of the areas of the circles

Add the areas of the three circles:

$$ \text{Total Area}{\text{circles}} = \text{Area}{\text{circle}1} + \text{Area}{\text{circle}2} + \text{Area}{\text{circle}_3} $$

$$ \text{Total Area}_{\text{circles}} = 113.04 + 254.34 + 452.16 = 819.54 , \text{in}^2 $$

4. Calculate the area of the shaded region

Finally, subtract the total area of the circles from the area of the rectangle:

$$ \text{Area}{\text{shaded}} = \text{Area}{\text{rectangle}} - \text{Total Area}_{\text{circles}} $$

$$ \text{Area}_{\text{shaded}} = 216 - 819.54 = -603.54 , \text{in}^2 $$

This negative area indicates an error in assumptions or dimensions, as a shaded area cannot be negative.