A rectangular window measures 54 inches by 24 inches. There is an a = 16-inch wiper blade attached by a b = 4-inch arm at the center of the base of the window. The wiper arm rotates 120°.
a. Find the area, in sq inches, of the window that is wiped. (Round your answer to two decimal places.)
b. Approximate the percentage of the window's area that is wiped by the blade. (Round your answer to one decimal place.)

Given the problem:

- The window has dimensions 54 inches by 24 inches.
- There is a wiper with a blade of length \( a = 16 \) inches.
- The arm of the wiper is of length \( b = 4 \) inches.
- The wiper arm rotates through an angle of 120°.

### Part (a): Finding the area of the window that is wiped

The wiper sweeps an arc of a circle centered at the point where the arm is attached. The total effective length of the wiper is \( a + b = 16 + 4 = 20 \) inches.

The angle of rotation is \( 120^\circ \), which is \( \frac{120}{360} = \frac{1}{3} \) of a full circle.

The area swept by the wiper blade can be calculated as the area of a sector of a circle:
\[
A_{\text{wiped}} = \frac{\theta}{360^\circ} \cdot \pi r^2
\]
where \( r = 20 \) inches and \( \theta = 120^\circ \).

Substituting the values:
\[
A_{\text{wiped}} = \frac{120^\circ}{360^\circ} \cdot \pi (20)^2
\]
Now, let's calculate this area.

### Part (b): Approximate the percentage of the window's area that is wiped by the blade

To find the percentage of the window's area that is wiped, we first need to find the total area of the window:
\[
A_{\text{window}} = 54 \times 24 \text{ square inches}
\]
Then, the percentage of the window wiped is:
\[
\text{Percentage wiped} = \frac{A_{\text{wiped}}}{A_{\text{window}}} \times 100
\]

Let's calculate both parts.import math

# Given values
a = 16  # Length of the wiper blade in inches
b = 4   # Length of the arm in inches
theta = 120  # Angle in degrees
radius = a + b  # Effective radius
window_width = 54  # Width of the window in inches
window_height = 24  # Height of the window in inches

# Calculate the area wiped by the wiper blade (sector area of a circle)
sector_area = (theta / 360) * math.pi * radius**2

# Calculate the total area of the window
window_area = window_width * window_height

# Calculate the percentage of the window that is wiped
percentage_wiped = (sector_area / window_area) * 100

# Round the results to two decimal places for area and one decimal place for percentage
sector_area_rounded = round(sector_area, 2)
percentage_wiped_rounded = round(percentage_wiped, 1)

sector_area_rounded, percentage_wiped_rounded### Solutions:
**(a)** The area of the window that is wiped by the wiper blade is approximately **418.88 square inches**.

**(b)** The percentage of the window's area that is wiped by the blade is approximately **32.3%**.

Would you like more details on any part of the calculations?

Here are 5 related questions for further exploration:
1. What would the area wiped be if the wiper rotated through a 90° angle instead of 120°?
2. How would the wiped area change if the length of the wiper blade was increased to 20 inches?
3. What would happen to the percentage wiped if the window dimensions were doubled?
4. How would the wiped area change if the arm length was reduced to 2 inches?
5. What would be the effect on the wiped area if the window were circular with a diameter of 54 inches?

**Tip:** The area of a sector depends both on the radius and the angle of rotation, so changing either will affect the result significantly.

Learn how to calculate the area wiped by a wiper blade rotating 120° on a rectangular window measuring 54 inches by 24 inches. Includes step-by-step calculations and results: area wiped approximately 418.88 square inches and percentage of window's area wiped approximately 32.3%.

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