From a rectangle ABCD of area 768 sq cm, a semicircular part with diameter AB and area 72π sq cm is removed. The perimeter of the leftover portion, in cm, is
- 80+16π
- 86+8π
- 82+24π
- 88+12π
The Correct Option is D
Solution and Explanation
We are given a semicircle with diameter \( AB \), and rectangle \( ABCD \). Let's go step-by-step to determine the dimensions and total perimeter.
Step 1: Area of the Semicircle
The formula for the area of a semicircle with diameter \( AB \) is:
\[ \text{Area}_{\text{semi}} = \frac{1}{2} \times \pi \times \left( \frac{AB}{2} \right)^2 \]
This is given to be \( 72\pi \). Equating:
\[ \frac{1}{2} \times \pi \times \left( \frac{AB}{2} \right)^2 = 72\pi \]
Cancel \( \pi \) from both sides and simplify:
\[ \frac{1}{2} \times \left( \frac{AB^2}{4} \right) = 72 \quad \Rightarrow \quad \frac{AB^2}{8} = 72 \]
\[ AB^2 = 576 \quad \Rightarrow \quad AB = \sqrt{576} = 24 \, \text{cm} \]
Step 2: Area of the Rectangle ABCD
The area of rectangle \( ABCD \) is given as 768 cm². Since \( AB = 24 \), and \( BC \) is the adjacent side:
\[ AB \times BC = 768 \quad \Rightarrow \quad 24 \times BC = 768 \]
\[ BC = \frac{768}{24} = 32 \, \text{cm} \]
Step 3: Perimeter of the Remaining Shape
The shape consists of three rectangle sides and one semicircular arc (along \( AB \)). The sides contributing to the perimeter are:
- \( AD = BC = 32 \)
- \( DC = AB = 24 \)
- \( CB = BC = 32 \)
- Arc \( AB \) = semicircular arc = \( \frac{1}{2} \times \pi \times AB = \frac{1}{2} \pi \times 24 = 12\pi \)
Therefore, the total perimeter is:
\[ \text{Perimeter} = 32 + 24 + 32 + 12\pi = 88 + 12\pi \, \text{cm} \]
Final Answer:
\[ \boxed{\text{Perimeter} = 88 + 12\pi \, \text{cm}} \]
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