A window cleaner has several windows to clean for a local business. What is the area of the circular window to the nearest hundredth of a square foot?

Steps to Solve

1. Determine the Radius The diameter of the circular window is given as ( 3 \frac{1}{2} ) feet. First, convert this mixed number into an improper fraction: $$ 3 \frac{1}{2} = \frac{7}{2} \text{ feet} $$ To find the radius, divide the diameter by 2: $$ \text{Radius} = \frac{7}{2} \cdot \frac{1}{2} = \frac{7}{4} \text{ feet} $$

2. Use the Area Formula The formula for the area ( A ) of a circle is given by: $$ A = \pi r^2 $$ Substituting the radius ( r = \frac{7}{4} ): $$ A = \pi \left(\frac{7}{4}\right)^2 $$

3. Calculate ( r^2 ) Now, compute ( r^2 ): $$ \left(\frac{7}{4}\right)^2 = \frac{49}{16} $$

4. Calculate the Area Replacing ( r^2 ) in the area formula gives: $$ A = \pi \cdot \frac{49}{16} $$ Using ( \pi \approx 3.14 ): $$ A \approx 3.14 \cdot \frac{49}{16} = 3.14 \cdot 3.0625 $$

5. Final Calculation Now calculate the above multiplication: $$ A \approx 9.615625 $$ Rounding to the nearest hundredth: $$ A \approx 9.62 \text{ square feet} $$