A light on the ground moving at 0.5 m/s approaches a man standing 4 m from a wall. How fast is the tip of the man’s shadow moving when the light is 10 m from the wall?

depends on the man's height, h. If the shadow's height is s and the light is x m from the wall, then using similar triangles, we have

x/s = (x-4)/h
x/(x-4) = s/h
-4/(x-4)^2 dx/dt = 1/h ds/dt

So, plugging your numbers,

-4/36 * -1/2 = 1/h ds/dt
ds/dt = h/18

To find the speed at which the tip of the man's shadow is moving, we can use the concept of similar triangles and the chain rule from calculus.

Let's assume that the height of the man is h and the length of his shadow is x. Since the light is on the ground, we can say that the height of the light source and the length of its shadow are negligible.

We are given that the light is moving at a constant rate of 0.5 m/s, which means that the rate of change of the distance from the light source to the wall is 0.5 m/s as well.

When the light source is 10 m from the wall, the distance from the man to the wall is 4 m. Therefore, the length of the shadow of the man is x = 4 m.

Let's use the variables t for time, L for the distance from the light to the wall, X for the distance from the man to the wall, and H for the height of the man.

Using the similar triangles formed by the man, his shadow, and the wall, we have the proportional relationship:

H / X = (H + h) / (X + x)

Taking the derivative of both sides with respect to time t, we get:

(dH/dt) / X = [(dH/dt) + (dh/dt)] / (X + x)

Since we are interested in finding the rate at which the tip of the shadow is moving, we want to solve for (dh/dt). Rearranging the equation, we have:

(dh/dt) = [(dH/dt) / X] * (X + x) - (dH/dt)

Substituting the given values, we have:

(dh/dt) = [(0.5 m/s) / 4 m] * (4 m + 10 m) - (0.5 m/s)

Simplifying the equation:

(dh/dt) = (0.5/4) * 14 - 0.5
= 0.875 - 0.5
= 0.375 m/s

So, the tip of the man's shadow is moving at a speed of 0.375 m/s when the light is 10 m from the wall.