A girl while walking diametrically across a semicircular playground takes 3 minutes less than if she had kept walking round the circular path from A to B If she walks 60 metres a minute, what is the diameter of the play ground

Options

A. 60 m
B. 48 m
C. 84 m
D. 315 m

Answer

To solve the problem, we need to find the diameter of the semicircular playground based on the information given. Let's break it down step by step.

### Step 1: Define the Variables
Let the diameter of the semicircular playground be \( D \) meters. The radius \( R \) will then be \( \frac{D}{2} \).

### Step 2: Calculate the Distance for Each Path
1. **Distance across the semicircle (diametrically)**: The distance when walking across the semicircle is equal to the diameter, which is \( D \) meters.
2. **Distance around the semicircle**: The circumference of a full circle is given by \( \pi D \). Therefore, the distance around the semicircle (which is half the circumference) is \( \frac{\pi D}{2} \) meters.

### Step 3: Calculate the Time Taken for Each Path
1. **Time taken to walk across the semicircle**:
\[ \text{Time across} = \frac{D}{60} \text{ minutes} \]
2. **Time taken to walk around the semicircle**:
\[ \text{Time around} = \frac{\frac{\pi D}{2}}{60} = \frac{\pi D}{120} \text{ minutes} \]

### Step 4: Set Up the Equation
According to the problem, the time taken to walk across the semicircle is 3 minutes less than the time taken to walk around it. Therefore, we can set up the equation:
\[ \frac{D}{60} = \frac{\pi D}{120} - 3 \]

### Step 5: Solve the Equation
1. Multiply through by 120 to eliminate the denominators:
\[ 2D = \pi D - 360 \]
2. Rearranging gives:
\[ \pi D - 2D = 360 \]
\[ D(\pi - 2) = 360 \]
3. Finally, solve for \( D \):
\[ D = \frac{360}{\pi - 2} \]

### Step 6: Calculate the Diameter
Using the approximate value of \( \pi \approx 3.14 \):
\[ D \approx \frac{360}{3.14 - 2} = \frac{360}{1.14} \approx 315.79 \text{ meters} \]

### Conclusion
The diameter of the playground is approximately \( 315.79 \) meters.

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