To find the rate at which the length of the man's shadow is decreasing, we need to use similar triangles.
Let's denote the length of the man's shadow on the building as x (in meters) and the distance at which the man is from the wall as y (in meters).
Since the light is on the ground, the height of the man, the distance from the light to the wall, and the length of his shadow on the building form a right-angled triangle. Thus, we have:
x / y = 2 / 12
Cross-multiplying, we get:
12x = 2y
Differentiating both sides with respect to time (t) using the Chain Rule, we have:
12(dx/dt) = 2(dy/dt)
Now, let's substitute the given values into the equation:
When the man is 4 meters from the building (y = 4), we need to find how fast the length of his shadow x is decreasing (dx/dt).
Using the given speed at which the man is walking (2.3 m/s), we can find how fast the man's distance to the wall is changing (dy/dt = -2.3 m/s).
Plugging in the values, we have:
12(dx/dt) = 2(-2.3)
Simplifying the equation, we get:
12(dx/dt) = -4.6
Now, let's solve for dx/dt:
(dx/dt) = -4.6 / 12
Simplifying, we find:
(dx/dt) β -0.383 m/s
Therefore, when the man is 4 meters from the building, the length of his shadow on the building is decreasing at a rate of approximately 0.383 m/s.