A person is running along a circular track of area ` 625 pi m ^(2) ( pi = 22 // 7)` (= 22/7) with a constant speed. Find the displacement in 15 seconds if he has to complete the race in 30 seconds.

Options

A. 200 m
B. 100 m
C. 25 m
D. 50 m

Explanation

To solve the problem step by step, we need to determine the displacement of a person running along a circular track after 15 seconds, given that the area of the track is \(625 \pi \, m^2\) and the total time to complete the race is 30 seconds.

### Step 1: Calculate the radius of the circular track
The area \(A\) of a circle is given by the formula:
\[ A = \pi R^2 \]
Given that the area is \(625 \pi \, m^2\), we can set up the equation:
\[ \pi R^2 = 625 \pi \]
Dividing both sides by \(\pi\):
\[ R^2 = 625 \]
Taking the square root of both sides:
\[ R = \sqrt{625} = 25 \, m \]

### Step 2: Determine the total distance of the circular track
The circumference \(C\) of a circle is given by:
\[ C = 2 \pi R \]
Substituting the value of \(R\):
\[ C = 2 \pi (25) = 50 \pi \, m \]
Using \(\pi = \frac{22}{7}\):
\[ C = 50 \times \frac{22}{7} = \frac{1100}{7} \, m \approx 157.14 \, m \]

### Step 3: Calculate the distance traveled in 15 seconds
Since the person completes the race in 30 seconds, in 15 seconds (which is half of 30 seconds), the distance traveled will be half of the total circumference:
\[ \text{Distance in 15 seconds} = \frac{1}{2} C = \frac{1}{2} (50 \pi) = 25 \pi \, m \]

### Step 4: Determine the displacement after 15 seconds
After 15 seconds, the person will have traveled half the circumference of the circle, which means they will be at the opposite side of the starting point. The displacement is the straight-line distance between the starting point (point A) and the point directly opposite (point B) on the circle.

The displacement is equal to the diameter of the circle:
\[ \text{Diameter} = 2R = 2 \times 25 = 50 \, m \]

### Final Answer
Thus, the displacement of the person after 15 seconds is:
\[ \text{Displacement} = 50 \, m \]

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