Question

A street light is at the top of a 15 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 along the ground when she is 30 ft from the base of the pole

Answer 1

10ft/sec

Explanation:

From the question, we say that both triangles are similar.

we are given;

Height of pole= 15ft

Height of the woman = 6ft

Speed = 6ft/sec

Let's make use of the triangle equation:

(y - x) ÷ y = 6 ÷ 15

Solving the equation, we have:

15(y-x) = 6y

= 15y - 15x = 6y

= 15y - 6y = 15x

= 9y = 15x

y = 5 / 3x

Let's differentiate both sides with respect to time,

Therefore we have:

dy/dt = 5/3

At period t, she walks away,

we now have:

dx/dt = 6ft/sec

Therefore

= dy/dt= 5/3 * 6ft/sec

dy/dt = 30/3

dy/dt = 10ft/second

The woman's distance from the pole won't count because only speed affects the movement of shadow.

Therefore the speed of her shadow moving along the ground when she is 30ft from the base of the pole is 10ft/sec.

Answer 2

`10(ft)/(s)`

Explanation:

in this diagram, x is the distance from the man to the pole, and y is the distance from the tip of the man's shadow to the pole. i assume the man and pole are standing straight up, which means the 2 triangles are similar.

by similarity,
`(y-x)/(y) =(6)/(15)`

15(y-x) =6y

15y-15x=6y

y=
`(5)/(3)x`

`(dy)/(dt) = (5)/(3) (dx)/(dt)` (differentiating both sides w.r.t time 't')

As you know
`(dx)/(y\dt)= 6(ft)/(s)` because the woman is walking that speed away from the pole.

to find how fast the tip of the shadow is moving i.e
`(dy)/(dt)`

`(dy)/(dt) = (5)/(3) . 6(ft)/(s) = (30)/(3) ft/s \\(dy)/(dt)= 10(ft)/(s)`

so, her distance from the pole if of no use since only her speed affects how fast her shadow moves.