A bright light on the ground illuminates a wall 12 meters away. A man walks from the light straight toward the building at a speed of 2.3 m/s. The man is 2 meters tall. When the man is 4 meters from the building, how fast is the length of his shadow on the building decreasing?

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) ≈ -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.

To find the rate at which the length of the man's shadow on the building is decreasing, we can use similar triangles.

Let's assume that the length of the man's shadow is represented by the variable "x". As the man walks towards the building, the distance between the man and the light source (which is also the distance between the man and the building) is decreasing. Let's represent this distance by the variable "d".

We know that the initial distance between the man and the building is 12 meters, and the rate at which he is walking towards the building is 2.3 m/s. Therefore, after a time "t" seconds, the distance between the man and the building can be represented as:

d = 12 - (2.3 * t)

Now, let's consider the similar triangles formed by the man, the light source, and the building. The height of the man (2 meters) and the length of his shadow (x) are proportional to the distance between the man and the building (d).

We can set up the following proportion:

2 / x = 12 / d

Substituting the expression we found for "d" earlier into the proportion:

2 / x = 12 / (12 - (2.3 * t))

Now we can solve for "x" by cross-multiplying:

2(12 - (2.3 * t)) = 12x

Expanding and rearranging the equation:

24 - 4.6t = 12x

Dividing by 12:

2 - 0.383t = x

Now, to find the rate at which the length of the man's shadow is decreasing (dx/dt), we differentiate both sides of the equation with respect to time "t":

(dx/dt) = -0.383

Therefore, the length of the man's shadow on the building is decreasing at a constant rate of 0.383 meters per second as he walks towards the building when he is 4 meters away.