Question - Calculate the Area of a Shaded Region in a Combined Shape
Solution:
The shaded area is made up of a rectangular part and a quarter of a circular sector (since we're given a right angle and the arc of a circle with center P). The rectangle has one side that is 7 m and shares the other side with the radius of the sector, which is 7 m. The area of the rectangle \( A_{rectangle} \) is:
\[ A_{rectangle} = \text{length} \times \text{width} \]
\[ A_{rectangle} = 7 \times 7 \]
\[ A_{rectangle} = 49 \ m^2 \]
The area of the circular sector \( A_{sector} \) is a quarter of the area of a circle with radius 7 m:
\[ A_{sector} = \frac{1}{4} \pi r^2 \]
\[ A_{sector} = \frac{1}{4} \pi (7)^2 \]
\[ A_{sector} = \frac{1}{4} \pi (49) \]
\[ A_{sector} = 12.25 \pi \ m^2 \]
Therefore, the total shaded area \( A \) is the sum of the rectangle area and the sector area:
\[ A = A_{rectangle} + A_{sector} \]
\[ A = 49 + 12.25 \pi \]
\[ A \approx 49 + 38.48 \]
\[ A \approx 87.48 \ m^2 \]
So, the area of the shaded region is approximately 87.48 square meters.
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