The person is walking at a constant rate, the length of their shadow is changing at a decreasing rate.
No, the length of the person's shadow is not changing at a constant rate.
Although the person is moving at a constant rate, the rate of change of the shadow's length is dependent on the distance between the person and the light pole.
As the person moves further away, the rate of change of the shadow length decreases.
Here's why:
Similar triangles: The situation can be modeled by two similar triangles: one with the person and their shadow, and the other with the light pole and the shadow.
The ratio of corresponding sides in similar triangles is constant.
Changing distance: As the person walks away, the distance between the person and the light pole increases. This, in turn, increases the length of the shadow.
Decreasing rate of change: However, the rate of change of the shadow length is not constant.
It decreases as the person moves further away. This is because the angle between the person's path and the shadow shrinks as the person moves away.
Here's a mathematical way to understand this:
Let:
h be the height of the person (1.77 meters)
l be the length of the shadow
d be the distance between the person and the light pole (initially 0 meters)
θ be the angle between the person's path and the shadow
We can use the similar triangles relationship:
l / h = d / (d + 3.26)
Differentiating both sides with respect to time (t) gives:
dl/dt = (h * d * dθ/dt) / (d + 3.26)^2
As you can see, the rate of change of the shadow length (dl/dt) depends on the distance (d), which is increasing with time.
This means that dl/dt is not constant and will decrease as the person walks further away from the light pole.
Therefore, while the person is walking at a constant rate, the length of their shadow is changing at a decreasing rate.