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
The streelight is
His distance from the streetlight is: .
. . (decreasing)
The length of his shadow is: .
. . We want:
From the similar triangles: .
Differentiate with respect to time: .
Therefore: . 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
The streelight is
His distance from the streetlight is: .
. . (decreasing)
The distance of the tip of his shadow to the streetlight is:
From the similar triangles: .
Differentiate with respect to time: .
Therefore: . 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 moving towards his feet ?
. . We found this to be: . ft/sec.
In part (b), they ask for the rate of change of point relative to the world.
Relative to the streetlight, the distance is
. . We found that: . 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.