What is the area of the colored region if the radii of the six circles are 1, 2, 3, 4, 5, and 10 centimeters? (Use 3.14 for π. Round to the nearest tenth if necessary.)

Steps to Solve

1. Calculate the area of the largest circle The radius of the largest circle is 10 cm. The area of a circle is given by the formula $A = \pi r^2$. We're given that $\pi = 3.14$. So the area of the largest circle is: $A_{large} = 3.14 \times (10)^2 = 3.14 \times 100 = 314 \text{ cm}^2$

2. Calculate the areas of the smaller circles The radii of the smaller circles are 1, 2, 3, 4, and 5 cm. We need to calculate their individual areas and then sum them up.

$A_1 = 3.14 \times (1)^2 = 3.14 \text{ cm}^2$ $A_2 = 3.14 \times (2)^2 = 3.14 \times 4 = 12.56 \text{ cm}^2$ $A_3 = 3.14 \times (3)^2 = 3.14 \times 9 = 28.26 \text{ cm}^2$ $A_4 = 3.14 \times (4)^2 = 3.14 \times 16 = 50.24 \text{ cm}^2$ $A_5 = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \text{ cm}^2$

3. Calculate the total area of the smaller circles Sum the areas of the smaller circles: $A_{small} = A_1 + A_2 + A_3 + A_4 + A_5 = 3.14 + 12.56 + 28.26 + 50.24 + 78.5 = 172.7 \text{ cm}^2$

4. Calculate the area of the colored region Subtract the total area of the smaller circles from the area of the largest circle: $A_{colored} = A_{large} - A_{small} = 314 - 172.7 = 141.3 \text{ cm}^2$