A man who is 1.6 m tall is walking on the road at a constant speed of 1 m/s. There is (only) one lamp placed 3.2 m above the road. At a specific moment the man was just under the lamp. What will be the rate at which the length of his shadow will be changing after: a) 2 sec b) 5 sec?

Question

A man who is 1.6 m tall is walking on the road at a constant speed of 1 m/s. There is (only) one lamp placed 3.2 m above the road. At a specific moment the man was just under the lamp. What will be the rate at which the length of his shadow will be changing after: a) 2 sec b) 5 sec?

Answer 1

To find the rate at which the length of the man's shadow is changing at a given moment, we need to make use of similar triangles. Let's denote the length of the man's shadow as "s" and the distance from him to the lamp as "d".

Since the man's height is 1.6 m and the lamp is placed 3.2 m above the road, the total length from the lamp to the man is 1.6 m + 3.2 m = 4.8 m.

At any given moment, the height of the man's shadow (h) and the height of the man (1.6 m) will be proportional to the distance of the man from the lamp (d) and the length of his shadow (s), respectively. This can be represented by the equation:

h/d = 1.6/s

Now, let's differentiate both sides of the equation with respect to time (t):

dh/dt = (1.6 ds/dt)/s^2

We know that the man is walking at a constant speed of 1 m/s, so the rate at which the distance (d) is changing is equal to 1 m/s. Therefore, ds/dt = 1 m/s.

Now, we can substitute the values into the equation:

dh/dt = (1.6 * 1) / s^2

Simplifying further:

dh/dt = 1.6 / s^2

a) After 2 seconds:
To find the rate at which the length of the man's shadow is changing after 2 seconds, we need to find the length of the shadow (s) at that time. Since the man walks at a constant speed of 1 m/s, after 2 seconds, the distance traveled is 2 m.

Using the equation s = d + h, where d is the distance from the lamp to the man and h is the man's height:

s = d + h
s = 2 + 1.6
s = 3.6 m

Now substitute the value of "s" into the equation dh/dt = 1.6 / s^2:

dh/dt = 1.6 / (3.6)^2
dh/dt = 1.6 / 12.96
dh/dt = 0.1235 m/s

Therefore, the rate at which the length of the man's shadow is changing after 2 seconds is 0.1235 m/s.

b) After 5 seconds:
Similarly, after 5 seconds, the distance traveled is 5 m.

Using the same equation as before:

s = d + h
s = 5 + 1.6
s = 6.6 m

Now substitute the value of "s" into the equation dh/dt = 1.6 / s^2:
dh/dt = 1.6 / (6.6)^2
dh/dt = 1.6 / 43.56
dh/dt = 0.0368 m/s

Therefore, the rate at which the length of the man's shadow is changing after 5 seconds is 0.0368 m/s.