Question

1. a windshield wiper blade is 18 inches long and rotates through an angle of 110°.

a) what is the length of the arc traced by the tip of the wiper blade?
b) what is the area of the windshield cleaned by one sweep of the wiper?
round both answers to the nearest tenth

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) is \( s = r\theta \), where \( r \) is the radius (length of the wiper blade, 18 inches) and \( \theta \) is the central angle in radians. First, convert \( 110^\circ \) to radians: \( \theta = 110^\circ \times \frac{\pi}{180^\circ} = \frac{11\pi}{18} \) radians.

Step2: Calculate arc length

Substitute \( r = 18 \) and \( \theta = \frac{11\pi}{18} \) into the arc length formula: \( s = 18 \times \frac{11\pi}{18} = 11\pi \approx 34.6 \) inches (rounded to the nearest tenth).

Step3: Recall sector area formula

The formula for the area of a sector \( A \) is \( A = \frac{1}{2}r^2\theta \), where \( r = 18 \) and \( \theta = \frac{11\pi}{18} \) radians.

Step4: Calculate sector area

Substitute the values into the formula: \( A = \frac{1}{2} \times 18^2 \times \frac{11\pi}{18} = \frac{1}{2} \times 18 \times 11\pi = 99\pi \approx 311.0 \) square inches (rounded to the nearest tenth).

Answer:

s:
a) The length of the arc is approximately \( \boldsymbol{34.6} \) inches.
b) The area of the windshield cleaned is approximately \( \boldsymbol{311.0} \) square inches.