Question

A sprinter starts running on a circular path of radius \( r \) metres. Her average speed (in metres/minute) is \( \pi r \) during the first 30 seconds, \( \pi r/2 \) during next one minute, \( \pi r/4 \) during next 2 minutes, \( \pi r/8 \) during next 4 minutes, and so on. What is the ratio of the time taken for the nth round to that for the previous round?

• A. 4
• B. 16
• C. 8
• D. 32

Hint:

Compute the distance covered in each successive time block; notice they are all equal, so the number of blocks per round is fixed.

Answer 1

The Correct Option is B

Step 1: Distance in each block:
Block 1: 0.5 min at \( \pi r \) m/min gives \( \pi r/2 \) m. Block 2: 1 min at \( \pi r/2 \) gives \( \pi r/2 \). Block 3: 2 min at \( \pi r/4 \) gives \( \pi r/2 \). Block k: duration \( 2^{k-2} \) min, speed \( \pi r/2^{k-1} \), distance \( \pi r/2 \).

Step 2: Blocks per round:
Circumference = \( 2\pi r \) = 4 x (\( \pi r/2 \)). So every round consists of exactly 4 blocks; round n consists of blocks 4n-3, 4n-2, 4n-1, 4n.

Step 3: Time of a round:
Time of block k = \( 2^{k-2} \) min. Time of round n = \( 2^{4n-5}(1+2+4+8)=15\cdot2^{4n-5} \) min.

Step 4: Ratio:
\( \dfrac{\text{time of round } n}{\text{time of round } n-1}=\dfrac{15\cdot2^{4n-5}}{15\cdot2^{4n-9}}=2^4=16 \).

Final Answer:
The correct option is (2): 16