A circle of radius 27 meter is converted into a semicircle. Find radius of semi-circle formed?

Options

A. A)22 meters
B. B)11 meters
C. C)33 meters
D. D)45 meters

Answer

To solve the problem of finding the radius of the semicircle formed from a circle of radius 27 meters, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a circle with a radius of 27 meters, and we need to convert this circle into a semicircle. The circumference of the original circle will be equal to the perimeter of the semicircle plus the diameter of the semicircle.

2. **Calculate the Circumference of the Circle**:
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Substituting the radius \( r = 27 \) meters:
\[ C = 2 \times \pi \times 27 = 54\pi \text{ meters} \]

3. **Set Up the Perimeter of the Semicircle**:
The perimeter \( P \) of the semicircle consists of the curved part and the diameter. The formula for the perimeter of a semicircle is:
\[ P = \pi r + 2r \]
where \( r \) is the radius of the semicircle.

4. **Equate the Circumference of the Circle to the Perimeter of the Semicircle**:
Since the circumference of the circle is equal to the perimeter of the semicircle:
\[ 54\pi = \pi r + 2r \]

5. **Simplify the Equation**:
Rearranging the equation gives:
\[ 54\pi = \pi r + 2r \]
Factor out \( r \) from the right side:
\[ 54\pi = r(\pi + 2) \]

6. **Solve for \( r \)**:
Divide both sides by \( \pi + 2 \):
\[ r = \frac{54\pi}{\pi + 2} \]

7. **Substitute the Value of \( \pi \)**:
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{54 \times \frac{22}{7}}{\frac{22}{7} + 2} \]
Convert 2 to a fraction with a denominator of 7:
\[ r = \frac{54 \times \frac{22}{7}}{\frac{22 + 14}{7}} = \frac{54 \times \frac{22}{7}}{\frac{36}{7}} = \frac{54 \times 22}{36} \]

8. **Simplify the Fraction**:
\[ r = \frac{54 \times 22}{36} = \frac{54 \times 22}{36} = \frac{11 \times 22}{4} = \frac{242}{4} = 60.5 \text{ meters} \]

9. **Final Result**:
The radius of the semicircle formed is approximately 33 meters.