Let's first draw a diagram to visualize the situation.

We have a spotlight on the ground, which we can represent as the point S. The wall is represented by the line segment WX. The person is represented by the point P and the length of their shadow is represented by the line segment PS.

We are given that the spotlight is located 30 meters away from the wall, so we can label this distance as 30.

Since the person is walking towards the wall at a constant rate of 2 meters per second, we can represent their movement with a line segment PQ. The person is 24 meters from the wall, so we can label this distance as 24.

We are asked to find how fast the length of the person's shadow is changing when they are 24 meters from the wall. In other words, we want to find d(PS)/dt when P is 24.

Using similar triangles, we can express the length of the person's shadow in terms of the distance they are from the wall.

Let's label the length of the person's shadow as x. By similar triangles, we have the following proportion:

x/24 = (x + 30)/30

Simplifying this proportion, we get:

30x = 24(x + 30)

30x = 24x + 720

6x = 720

x = 120

So, when the person is 24 meters from the wall, the length of their shadow is 120 meters.

Now, let's differentiate both sides of the proportion with respect to time t.

d(x/24)/dt = d((x + 30)/30)/dt

Simplifying, we get:

(1/24) * dx/dt = (1/30) * d(x + 30)/dt

Solving for dx/dt, we have:

dx/dt = (1/24) * (1/30) * d(x + 30)/dt

dx/dt = (1/720) * d(x + 30)/dt

Substituting x = 120, we can find d(x + 30)/dt.

d(x + 30)/dt = d(120 + 30)/dt = d(150)/dt = 0

Therefore, dx/dt = (1/720) * 0 = 0.

So, when the person is 24 meters from the wall, the length of their shadow is not changing.

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