Question

A street light is mounted at the top of a 15 foot-tall pole. A man 6ft tall walks away from the pole with the speed of 5ft/s along a straight path. How fast is the tip of his shadow moving when he is 40ft from the pole? ft/s

Answer 1

The tip of the man's shadow is moving at a rate of 5ft/s when he is 40ft from the pole.

Step-by-step explanation:

To solve this problem, we can use similar triangles. Let's call the length of the shadow x. Since the man is 6ft tall and the pole is 15ft tall, we have the following proportion:

(6ft) / (x ft) = (15ft) / ((x + 5ft) ft)

Cross multiplying and solving for x gives us: (15ft)(x ft) = (6ft)((x + 5ft) ft)

Simplifying further, we get: 15x = 6(x + 5)

Now, we can solve for x:

15x = 6x + 30

9x = 30

x = 30/9

x = 3.33ft

The rate at which the tip of the shadow is moving can be found by taking the derivative of the equation for x with respect to time: dx/dt = d/dt(3.33ft)

Since the man is walking away from the pole with a speed of 5ft/s, we can substitute that value into the derivative: dx/dt = 5ft/s

Therefore, the tip of his shadow is moving at a rate of 5ft/s when he is 40ft from the pole.