A man 2 m tall is walking away from a lamppost which is 6m tall at a rate of 2 m/s. Find the rate of change of a)the tip of the shadow b) the length of his shadow
Call distance of man from lamp post x and distance of shadow tip from lamp post y.
Then by similar triangles:
(y-x)/2 = y/6
6 y - 6 x = 2 y
y = 1.5 x
we know dx/dt = 2
dy/dt = 1.5 * dx/dt = 3
dy/dt - dx/dt = speed of tip = 3-2 = 1
To find the rate of change of the tip of the shadow, we need to consider similar triangles formed by the man, the lamppost, and their shadows.
Let's define the following variables:
x = distance of the man from the lamppost
y = length of the shadow
a) Rate of change of the tip of the shadow:
We have a right triangle formed by the 6m lamppost, the 2m man, and the shadow. Since these triangles are similar, we can set up the following proportion:
(Length of the man's shadow) / (Length of the man) = (Length of the lamppost's shadow) / (Length of the lamppost)
y / 2 = (y + x) / 6
Now, we can solve this equation for y:
6y = 2(y + x)
6y = 2y + 2x
4y = 2x
y = 0.5x
To find the rate of change of the tip of the shadow, we need to differentiate y with respect to time (t), since both x and y are changing with time.
Differentiate both sides of the equation with respect to t:
d/dt (y) = d/dt (0.5x)
(dy/dt) = 0.5(dx/dt)
Since dx/dt is given as 2 m/s, we can substitute it into the equation:
(dy/dt) = 0.5 * 2
(dy/dt) = 1 m/s
Therefore, the rate of change of the tip of the shadow is 1 m/s.
b) Rate of change of the length of the shadow:
To find the rate of change of the length of the shadow, we need to differentiate y with respect to time (t).
Differentiate both sides of the equation with respect to t:
d/dt (y) = d/dt (0.5x)
(dy/dt) = 0.5(dx/dt)
Since dx/dt is given as 2 m/s, we can substitute it into the equation:
(dy/dt) = 0.5 * 2
(dy/dt) = 1 m/s
Therefore, the rate of change of the length of the shadow is also 1 m/s.