A takes 4 min 54 sec to run 1 km while B takes 5 min in this run. How many meters ahead A should B stand at the start so that both reach together at the end point of the race, then... A takes 4 min 54 sec to run 1 km while B takes 5 min in this run. How many meters ahead A should B stand at the start so that both reach together at the end point of the race, then the distance from the starting point to the end point is how far is? Read more

Steps to Solve

1. Calculate A and B's Speeds

To find how far A should be behind B, we first calculate their speeds.

A's time to run 1 km is 4 minutes 54 seconds, which is: $$ 4 , \text{min} + \frac{54 , \text{sec}}{60} = 4.9 , \text{min} $$

A's speed: $$\text{Speed of A} = \frac{1 , \text{km}}{4.9 , \text{min}} = \frac{1000 , \text{m}}{294 , \text{sec}} \approx 3.4 , \text{m/s} $$

B's time to run 1 km is 5 minutes (300 seconds). So, B's speed: $$\text{Speed of B} = \frac{1 , \text{km}}{5 , \text{min}} = \frac{1000 , \text{m}}{300 , \text{sec}} \approx 3.33 , \text{m/s} $$

2. Determine the Time Taken for the End of the Race

Since both A and B run 1 km, the time taken by both will depend on their speeds. Since A runs faster, we know the time it takes for A to reach 1 km is about 294 seconds.

3. Find the Time until they meet

Let ( d ) be the distance A should be behind B. They will meet when they cover the same time. The time taken by A to reach 1 km is equal to the distance B covers in this time: $$ t_B = t_A = 294 , \text{sec} $$ $$ d + \text{distance covered by A} = \text{distance covered by B} $$

The distance covered by B is: $$ d + v_A \cdot t = v_B \cdot t $$ $$ d + 3.4 \cdot 294 = 3.33 \cdot 294 $$

Rearranging gives us: $$ d = 3.33 \cdot 294 - 3.4 \cdot 294 $$

4. Calculate the Final Distance

Now we can plug in the values and calculate ( d ): $$ d = 3.33 \cdot 294 - 3.4 \cdot 294 $$ $$ = (3.33 - 3.4) \cdot 294 $$ $$ = -0.07 \cdot 294 \approx -20.58 $$

Therefore, the distance A should start behind B is approximately 20.58 meters.