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. Convert the diameter to a radius

   The radius ( r ) is half of the diameter. The given diameter is ( 3 \frac{1}{2} ) feet, which can be expressed as an improper fraction: $$ 3 \frac{1}{2} = \frac{7}{2} \text{ feet} $$ Thus, the radius is: $$ r = \frac{7}{4} \text{ feet} $$

2. Use the area formula for a circle

   The area ( A ) of a circle is calculated using the formula: $$ A = \pi r^2 $$

3. Substitute the radius into the formula

   Substituting the radius we found: $$ A = \pi \left( \frac{7}{4} \right)^2 $$

4. Calculate the area

   First, calculate ( \left( \frac{7}{4} \right)^2 ): $$ \left( \frac{7}{4} \right)^2 = \frac{49}{16} $$ Now substitute this back into the area formula: $$ A = \pi \left( \frac{49}{16} \right) $$

5. Approximate the area using ( \pi \approx 3.14 )

   Thus, $$ A \approx 3.14 \times \frac{49}{16} $$

6. Perform the multiplication

   Calculate ( 3.14 \times \frac{49}{16} ): $$ A \approx \frac{3.14 \times 49}{16} $$ First, find ( 3.14 \times 49 ): $$ 3.14 \times 49 \approx 153.86 $$ Then divide by 16: $$ A \approx \frac{153.86}{16} \approx 9.6175 $$

7. Round to the nearest hundredth

   Rounding ( 9.6175 ) to the nearest hundredth gives: $$ A \approx 9.62 \text{ square feet} $$