You are a lifeguard and spot a drowning child 50 meters along the shore and 40 meters from the shore to the child. You run along the shore and for a while and then jump into the water and swim from there directly to child. You can run at a rate of 4 meters per second and swim at a rate of 1.1 meters per second. How far along the shore should you run before jumping into the water in order to save the child? Round your answer to three decimal places.

38.559 meters

Step-by-step explanation:

Let the distance traveled along shore before the jump be "x"

meters

time take during run: speed = distance / time therefore time = distance / speed, i. e.:

x / 4

Now the distances are:

40

50 - x

a right triangle is formed so it is possible to use Pythagoras:

d ^ 2 = 40 ^ 2 + (50 - x) ^ 2

d = [40 ^ 2 + (50 - x) ^ 2] ^ (1/2)

time taken during swin:

[40 ^ 2 + (50 - x) ^ 2] ^ (1/2) / 1.1

total time:

x / 4 + [40 ^ 2 + (50 - x) ^ 2] ^ (1/2) / 1.1

we derive and we are left:

1/4 - (1 / 1.1) * (50-x) / [40 ^ 2 + (50 - x) ^ 2] ^ (1/2)

we equal 0:

1/4 - (1 / 1.1) * (50-x) / [40 ^ 2 + (50 - x) ^ 2] ^ (1/2) = 0

1.1 * [40 ^ 2 + (50 - x) ^ 2] ^ (1/2) = 4 * (50 - x)

we square both sides:

1.21 * (40 ^ 2 + (50 - x) ^ 2 = 16 * (50 - x) ^ 2

1936 + 1.21 * (50-x ^ 2) = 16 * (50 - x) ^ 2

1936 = 14.79 * (50 - x) ^ 2

(50 - x) ^ 2 = 1936 / 14.79

(50 - x) ^ 2 = 130.9

50 - x = + - 11.441

x = 50 + - 11.441

x1 = 50 + 11.441 = 61.441

x2 = 50 - 11.441 = 38.559

x1 cannot be because it is greater than 50 - x, the result would be negative, therefore the answer is x = 38.559 meters