A lemming runs from point A to a cliff at B at 4 m/s, jumps over the edge and falls to point C at an average speed of 25 m/s. The total distance from A to C is 500 m, and the total time taken is 41 seconds. Find the height BC of the cliff.

1. Problem Description

A lemming runs from point A to a cliff at B at 4 m/s, jumps over the edge and falls to point C at an average speed of 25 m/s. The total distance from A to C is 500 m, and the total time taken is 41 seconds. Find the height BC of the cliff.

2. Solution Steps

Let $d_1$ be the distance from A to B, and $t_1$ be the time taken to run from A to B.

Let $d_2$ be the distance from B to C, and $t_2$ be the time taken to fall from B to C.

We know that:

$d_1 + d_2 = 500$

$t_1 + t_2 = 41$

We also know that:

$d_1 = 4t_1$

$d_2 = 25t_2$

Substitute these into the first two equations:

$4t_1 + 25t_2 = 500$

$t_1 + t_2 = 41$

From the second equation, we can express $t_1$ in terms of $t_2$:

$t_1 = 41 - t_2$

Substitute this into the first equation:

$4(41 - t_2) + 25t_2 = 500$

$164 - 4t_2 + 25t_2 = 500$

$21t_2 = 500 - 164$

$21t_2 = 336$

$t_2 = \frac{336}{21} = 16$

Now we can find $d_2$:

$d_2 = 25t_2 = 25(16) = 400$

Since $d_2$ represents the height BC of the cliff, $BC = d_2$.

3. Final Answer

The height of the cliff BC is 400 m.