The perimeter (in metres) of a semicircle is numerically equal to its area (in square metres). The length of its diameter is (take `pi = (22)/(7)`)

Options

A. `3(6)/(11)` metres
B. `5(6)/(11)` metres
C. `6(6)/(11)` metres
D. `6(2)/(11)` metres

Answer

To solve the problem, we need to find the diameter of a semicircle where the perimeter is numerically equal to its area. Let's break this down step by step.

### Step 1: Understand the formulas for perimeter and area of a semicircle
The perimeter \( P \) of a semicircle is given by:
\[ P = \pi r + 2r \]
where \( r \) is the radius of the semicircle. The first term \( \pi r \) accounts for the curved part, and the second term \( 2r \) accounts for the diameter.

The area \( A \) of a semicircle is given by:
\[ A = \frac{1}{2} \pi r^2 \]

### Step 2: Set up the equation
According to the problem, the perimeter is equal to the area:
\[ \pi r + 2r = \frac{1}{2} \pi r^2 \]

### Step 3: Substitute \( \pi \) with \( \frac{22}{7} \)
Substituting \( \pi = \frac{22}{7} \) into the equation:
\[ \frac{22}{7} r + 2r = \frac{1}{2} \cdot \frac{22}{7} r^2 \]

### Step 4: Simplify the equation
To eliminate the fraction, multiply through by 14 (the least common multiple of the denominators):
\[ 14 \left( \frac{22}{7} r \right) + 14(2r) = 14 \left( \frac{1}{2} \cdot \frac{22}{7} r^2 \right) \]
This simplifies to:
\[ 44r + 28r = 7 \cdot 11r^2 \]
Combining like terms gives:
\[ 72r = 77r^2 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ 77r^2 - 72r = 0 \]
Factoring out \( r \):
\[ r(77r - 72) = 0 \]
This gives us two solutions:
1. \( r = 0 \) (not valid)
2. \( 77r - 72 = 0 \) which leads to \( r = \frac{72}{77} \)

### Step 6: Find the diameter
The diameter \( d \) is twice the radius:
\[ d = 2r = 2 \cdot \frac{72}{77} = \frac{144}{77} \]

### Step 7: Convert to a mixed number
To express \( \frac{144}{77} \) as a mixed number:
\[ \frac{144}{77} = 1 \frac{67}{77} \]

### Final Answer
The length of the diameter is \( \frac{144}{77} \) metres or approximately \( 1.87 \) metres.

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