A man 5.5 ft tall walks away from a lamp post 10 ft high at the rate of 8 ft/s. (a.) How fast does his shadow lengthen?
(b.) how fast does the tip of the shadow move?
make a sketch
let the distance of the man from the lamp-post be x
let the length of his shadow be y
by ratios:
5.5/y = 10/(x+y)
5.5x + 5.5y = 10y
5.5x = 4.5y
times 10
55x = 45y
11x = 9y
11 dx/dt = 9 dy/dt
11(8) = 9 dy/dt
dy/dt = 88/9 ft/s or 9 7/9 ft/s --> rate at which the shadow is lengthening.
How fast is the shadow moving??
d(x+y)/dt = dx/dt + dy/dt
= 8 + 88/9 = 160/9 ft/s or 17 7/9 ft/s
Thank You very much !! :D
why was it multiplied to 10?
To find the rate at which the man's shadow lengthens, we can set up a proportion between the man's height and the shadow's length. Let's denote the man's height as y and the shadow's length as x.
(a.) Rate of shadow lengthening:
Since the height of the man and the height of the lamp post form similar triangles with their corresponding sides, we can establish the following proportion:
y / x = (man's distance from the lamp post) / (lamp post's height).
Given that the man's distance from the lamp post is changing at a constant rate of 8 ft/s, we can differentiate both sides of the equation with respect to time (t) to find the rate of change of the shadow's length.
dy/dt / dx/dt = d(man's distance from the lamp post) / dt / lamp post's height.
Since dx/dt is given to be 8 ft/s, and the lamp post's height is 10 ft, we have:
dy/dt = (d(man's distance from the lamp post) / dt) * (10 / x).
(b.) Rate at which the tip of the shadow moves:
The distance from the tip of the shadow to the lamp post is the sum of the man's distance from the lamp post and the shadow's length (x).
Therefore, the rate at which the tip of the shadow moves is the sum of the rate at which the man's distance from the lamp post changes (8 ft/s) and the rate at which the shadow's length changes (dy/dt).
So, the rate at which the tip of the shadow moves is given by:
Rate = 8 ft/s + dy/dt.
Now, to obtain the value of dy/dt, we can calculate it using the formula derived in part (a).
This is the method to find the rates at which the man's shadow lengthens and the tip of the shadow moves.