Answer:
The area wiped by the wiper blade is [tex]177\pi[/tex]
Step-by-step explanation:
Given:
car's rear windshield wiper rotates = 140°
Total length of the wiper mechanism = 24 inches
Length of the wiper blade = 13 inches
To Find:
The area wiped by the wiper blade = ?
Solution:
The length of the wiper mechanism , [tex]r_1[/tex] = 24
The length of the wiper mechanism , [tex]r_2[/tex]= 13
Now arc radius
[tex]r_1-r_2[/tex]= 24-13 = 11
The wipe rotates 140 degrees
Step 1: converting degree into radians
[tex]\theta = 140 \times \frac{\pi}{180}[/tex]
[tex]\theta =\frac{7\pi}{9}[/tex]
Step 2: Finding the area of the sector of the circle with radius [tex]r_1[/tex]
The area of the sector
= [tex]\frac{1}{2}r_1 ^2\theta[/tex]
= [tex]\frac{1}{2}24^2\times \frac{7\pi}{9}[/tex]
= [tex]\frac{576}{2} \times \frac{7\pi}{9}[/tex]
= [tex]\frac{4032 \pi}{18}[/tex]---------------------------(1)
Step 2: Finding the area of the sector with radius [tex]r_2[/tex]
= [tex]\frac{1}{2}r_1 ^2\theta[/tex]
= [tex]\frac{1}{2}11^2\times \frac{7\pi}{9}[/tex]
= [tex]\frac{121}{2} \times \frac{7\pi}{9}[/tex]
= [tex]\frac{1183 \pi}{18}[/tex]---------------------------(2)
Step 3: Finding the area wiped by the wiper
The area wiped by the wiper = the area of the sector of the circle with radius [tex]r_1[/tex] - the area of the sector with radius [tex]r_2[/tex]
From(1) and(2)
The area wiped by the wiper = [tex]\frac{4032 \pi}{18}[/tex] - [tex]\frac{847\pi}{18}[/tex]
The area wiped by the wiper = [tex]\frac{3185\pi}{18}[/tex]
The area wiped by the wiper = [tex]177\pi[/tex]
