A person observed that he required `30 s` time to cross a circular ground a long its diameter than to cover it once along the boundary. If his speed was `30m//min`, then the radius of the circular ground is

Options

A. 10.5 cm
B. 3.5 m
C. 5.5 m
D. 7.5 m

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Understanding the Problem
We need to find the radius of a circular ground given that a person takes 30 seconds less time to cross the ground along its diameter than to cover it once along the boundary. The person's speed is given as 30 meters per minute.

### Step 2: Convert Speed to Meters per Second
The speed of the person is given in meters per minute. To convert this to meters per second:
\[ \text{Speed} = \frac{30 \text{ meters}}{1 \text{ minute}} = \frac{30}{60} \text{ meters/second} = 0.5 \text{ meters/second} \]

### Step 3: Formulate the Distances
1. **Distance along the diameter**: The diameter \(D\) of the circular ground is \(2r\), where \(r\) is the radius.
2. **Circumference (distance along the boundary)**: The circumference \(C\) of the circular ground is given by:
\[ C = 2\pi r \]

### Step 4: Calculate Time Taken for Each Path
1. **Time taken to cross along the diameter**:
\[ \text{Time}_{\text{diameter}} = \frac{\text{Distance}}{\text{Speed}} = \frac{2r}{0.5} = 4r \text{ seconds} \]
2. **Time taken to cover along the boundary**:
\[ \text{Time}_{\text{circumference}} = \frac{C}{\text{Speed}} = \frac{2\pi r}{0.5} = 4\pi r \text{ seconds} \]

### Step 5: Set Up the Equation
According to the problem, the time taken to cross along the diameter is 30 seconds less than the time taken to cover the boundary:
\[ 4r + 30 = 4\pi r \]

### Step 6: Rearranging the Equation
Rearranging the equation gives:
\[ 4\pi r - 4r = 30 \]
Factoring out \(4r\):
\[ 4r(\pi - 1) = 30 \]

### Step 7: Solve for \(r\)
Now, we can solve for \(r\):
\[ r = \frac{30}{4(\pi - 1)} = \frac{15}{2(\pi - 1)} \]

### Step 8: Substitute \(\pi\) and Calculate
Using \(\pi \approx \frac{22}{7}\):
\[ r = \frac{15}{2\left(\frac{22}{7} - 1\right)} = \frac{15}{2\left(\frac{22 - 7}{7}\right)} = \frac{15 \cdot 7}{2 \cdot 15} = \frac{7}{2} = 3.5 \text{ meters} \]

### Final Answer
The radius of the circular ground is \(3.5\) meters.

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