A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. (a) When he is 10 feet from the base of the light, at what rate is the tip of his shadow moving? (b) When he is 10 feet from the base of the light, at what rate is the length of his shadow changing?
Solution:
Given, the height of man = 6 feet
Height of light = 15 feet
Rate of walk by man, dx/dt = 5 feet per second.
a) we have to find the rate at which the tip of his shadow is moving when he is 10 feet from the base of the light.
By using ratio of similar triangles,
15/y = 6/(y - x)
15(y - x) = 6y
15y - 15x = 6y
15y - 6y = 15x
9y = 15x
Dividing by 3 on both sides,
3y = 5x
On differentiating both sides with respect to t,
3(dy/dt) = 5(dx/dt)
dy/dt = (5/3)(dx/dt)
dy/dt = (5/3)(5)
dy/dt = 30/3
dy/dt = 10 ft/s
Therefore, the rate at which the tip of his shadow is changing is 10 ft/s.
b) we have to find the rate at which the length of his shadow is changing when he is 10 feet from the base of the light.
Length of shadow, BD = y - x
Rate of change of shadow is calculated by differentiating length with respect to time.
d(BD)/dt = dy/dt - dx/dt
d(BD)/dt = 10 - 6
= 4 ft/s
Therefore, the rate at which the length of his shadow is changing is 4 ft/s.
Therefore,
(a) When he is 10 feet from the base of the light, the rate at which the tip of his shadow moves is 10ft/s (b) When he is 10 feet from the base of the light, the rate at which the length of his shadow changes is 4ft/s.
A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. (a) When he is 10 feet from the base of the light, at what rate is the tip of his shadow moving? (b) When he is 10 feet from the base of the light, at what rate is the length of his shadow changing?
Summary:
A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. (a) When he is 10 feet from the base of the light, the rate at which the tip of his shadow moves is 10ft/s (b) When he is 10 feet from the base of the light, the rate at which the length of his shadow changes is 4ft/s.