Hello, StarlitxSunshine!

I don't understand your proportion

. . but you got the right answer.

A man 6 ft tall is walking towards a streetlight 18 ft high at a rate of 3 ft/sec.

a) At what rate is his shadow length changing? Code:

. . C *
. . - | *
. . - | *
. . - | * A
. .18 | *
. . | | *
. . | 6| *
. . | | *
. . * - - - - - * - - - - - *
. . D p B s E

The man is $\displaystyle AB = 6.$

The streelight is $\displaystyle CD = 18.$

His distance from the streetlight is: .$\displaystyle p = BD.$

. . $\displaystyle \frac{dp}{dt} \:=\:3$ (decreasing)

The length of his shadow is: .$\displaystyle s = BE.$

. . We want: $\displaystyle \frac{dx}{dt}$

From the similar triangles: .$\displaystyle \frac{s}{6} \:=\:\frac{p+s}{18} \quad\Rightarrow\quad s \:=\:\tfrac{1}{2}p$

Differentiate with respect to time: .$\displaystyle \frac{ds}{dt} \:=\:\tfrac{1}{2}\,\frac{dp}{dt}$

Therefore: .$\displaystyle \frac{ds}{dt} \:=\:\tfrac{1}{2}(3) \:=\:\frac{3}{2}$ ft/sec.

b) How fast is the tip of his shadow moving?

I don't understand what they mean by the tip of his shadow.

Wouldn't it be moving at the same speed that the length is changing? No, this is a different situation.

Code:

. . C *
. . | *
. . | *
. . - | * A
. .18 | *
. . | | *
. . | 6| *
. . | | *
. . * - - - - - * - - - - - *
. . D p B x-p E
. . : - - - - - x - - - - - :

The man is $\displaystyle AB = 6.$

The streelight is $\displaystyle CD = 18.$

His distance from the streetlight is: .$\displaystyle p = BD.$

. . $\displaystyle \frac{dp}{dt} \:=\:3$ (decreasing)

The distance of the tip of his shadow to the streetlight is: $\displaystyle x \:=\:DE$

From the similar triangles: .$\displaystyle \frac{x-p}{6} \:=\:\frac{x}{18} \quad\Rightarrow\quad x \:=\:\tfrac{3}{2}p$

Differentiate with respect to time: .$\displaystyle \frac{dx}{dt} \:=\:\tfrac{3}{2}\frac{dp}{dt}$

Therefore: .$\displaystyle \frac{dx}{dt} \:=\:\tfrac{3}{2}(3) \:=\:\frac{9}{2}$ ft/sec.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Some explanation . . .

In part (a), they ask for the rate of change of the length of the shadow.

That is, how fast is the tip of his shadow $\displaystyle (E)$ moving towards his feet $\displaystyle (B)$?

. . We found this to be: .$\displaystyle 1\tfrac{1}{2}$ ft/sec.

In part (b), they ask for the rate of change of point $\displaystyle E$ *relative to the world.*

Relative to the streetlight, the distance is $\displaystyle x = DE.$

. . We found that: .$\displaystyle \frac{dx}{dt} \:=\:4\tfrac{1}{2}$ ft/sec.

To put it yet another way:

The tip of his shadow is moving towards his feet at 1½ ft/sec.

But his feet are moving at 3 ft/sec.

So, relative to world, the tip of his shadow is moving at 4½ ft/sec.