A street light is at the top of a 17 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 6 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 40 ft from the base of the pole?

Question

A street light is at the top of a 17 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 6 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 40 ft from the base of the pole?

👇

2 answers
(click or scroll down)

👇

Answer 1

Reiny answered
15 years ago

Draw a right-angled triangle ABC, B the right angle, and the vertical AB = 17 feet

Somewhere between B and C draw a vertical DE = 6 feet, the height of the woman

let BD = x and DC = y, the length of her shadow

given: dx/dt = 6 ft/s
find: dy/dt

(actually d(x+y)/dt, I'll come back to that later)

by similar triangles 6/y = 17/(x+y)
.
.
.
11y = 6x
then 11dy/dt = 6dx/dt
dy/dt = 6(6)/11 = 36/11 ft/s

so no matter where she is, the length of her shadow is increasing at 36/11 ft/s

but the woman herself is moving at 6 ft/sec

so her shadow is moving at 6+36/11 or
102/11 ft/s

Answer 2

Explain Bot answered
11 months ago

To determine how fast the tip of the woman's shadow is moving, we can use similar triangles. Let's break down the problem and establish some variables.

Let:
- The height of the pole be represented by "h" = 17 ft.
- The distance between the woman and the pole be represented by "x" = 40 ft.
- The height of the woman be represented by "y" = 6 ft.

We can set up a proportion between the height of the pole, the height of the woman, and the height of their respective shadows:

h + y x
------- = ---
h s

Where "s" represents the height of the shadow.

By cross-multiplying, we can rewrite the equation:

(h + y)s = hx

Now, we'll differentiate both sides of the equation with respect to time (t) and apply the Chain Rule:

(d/dt)[(h + y)s] = (d/dt)[hx]

(h + y) * ds/dt = h * dx/dt

We are given that dx/dt = 6 ft/sec (since the woman is moving away at a speed of 6 ft/sec). Substituting all the known values into the equation:

(17 + 6) * ds/dt = 17 * 6

23 * ds/dt = 102

Finally, we can solve for ds/dt, the rate at which the tip of the shadow is moving:

ds/dt = 102 / 23 ≈ 4.43 ft/sec

Therefore, the tip of the woman's shadow is moving at a rate of approximately 4.43 ft/sec when she is 40 ft from the base of the pole.