Question

A man riding a bicycle completes one lap of a circular field along its circumference at the speed of 14.4 km/hr in 1 minute 28 seconds. What is the area of the field?

• A. \(7958\) sq m
• B. \(9856\) sq m
• C. \(8842\) sq m
• D. \(8958\) sq m
• E. \(7856\) sq m

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A cyclist rides exactly once around the boundary (circumference) of a circular field.
We are given his steady riding speed in km/hr and the total time it took in minutes and seconds.
We need to calculate the total enclosed area of this circular field in square meters.
Step 2: Key Formula or Approach:
Since the options are in square meters, we must first convert the speed into meters per second (m/s) by multiplying by $\frac{5}{18}$.
Convert the time into purely seconds.
Calculate the total distance traveled using the formula: $\text{Distance} = \text{Speed} \times \text{Time}$.
This distance is exactly equal to the circumference of the circle, $C = 2\pi r$.
We will solve for the radius $r$ and then find the area using $\text{Area} = \pi r^2$.
Step 3: Detailed Explanation:

First, we convert the given speed of 14.4 km/hr into m/s.

Speed = $14.4 \times \frac{5}{18}$ m/s.

$14.4 \div 18 = 0.8$, so Speed = $0.8 \times 5 = 4$ m/s.

Next, we convert the given time of 1 minute 28 seconds into seconds.

Time = $60 \text{ seconds} + 28 \text{ seconds} = 88$ seconds.

Now we calculate the total distance covered in one lap, which is the circumference.

Circumference = Speed $\times$ Time = $4 \times 88 = 352$ meters.

The formula for circumference is $2\pi r$. So, $2\pi r = 352$.

Using $\pi = \frac{22}{7}$, we get $2 \times \frac{22}{7} \times r = 352$.

$\frac{44}{7} \times r = 352$.

Solving for $r$: $r = \frac{352 \times 7}{44}$.

Since $352 \div 44 = 8$, the radius $r = 8 \times 7 = 56$ meters.

Finally, we calculate the area of the circular field using the formula $\pi r^2$.

Area = $\frac{22}{7} \times 56 \times 56$.

Simplifying the expression: $22 \times 8 \times 56$.

$176 \times 56$.

We can compute this as $176 \times (50 + 6) = 8800 + 1056 = 9856$ sq.mt.

Step 4: Final Answer:
The area of the field is 9856 sq.mt.