High School

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A tip of a wiper moves through a distance of 176 cm in one sweep. Find the length of the wiper if it sweeps through an angle of 120°?

Answer :

To solve this problem, we need to understand the relationship between the arc length, the radius of the circle, and the angle through which the wiper sweeps.

We can use the formula for the arc length of a circle:

[tex]L = \frac{\theta}{360^\circ} \times 2\pi r[/tex]

Where:

  • [tex]L[/tex] is the arc length (in this case, the distance the wiper tip moves, which is 176 cm).
  • [tex]\theta[/tex] is the angle in degrees (given as 120°).
  • [tex]r[/tex] is the radius, which we need to find.

Given:
[tex]L = 176[/tex] cm,
[tex]\theta = 120^\circ[/tex]

Rearrange the formula to solve for [tex]r[/tex]:

[tex]r = \frac{L \times 360^\circ}{\theta \times 2\pi}[/tex]

Substitute the values:

[tex]r = \frac{176 \times 360}{120 \times 2\pi}[/tex]

Calculate the value:

  1. Calculate the products:
    [tex]176 \times 360 = 63360[/tex]
    [tex]120 \times 2\pi \approx 753.6[/tex] (using [tex]\pi \approx 3.14[/tex])

  2. Divide:
    [tex]r \approx \frac{63360}{753.6} \approx 84.05[/tex]

Thus, the length of the wiper is approximately 84.05 cm.

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