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 1.2 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?

Question

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 1.2 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?

Answer 1

As always, draw a diagram. Using similar triangles, the man's shadow height h when he is at a distance x from the light, is given by

2/x = h/12
or,
hx = 24
So, when the man is 4m from the wall (8m from the light) 8h=24, h=3

x dh/dt + h dx/dt = 0
Now plug in your values:

8 dh/dt + 3(1.2) = 0
dh/dt = -3.6/8 = -0.45 m/s

Answer 2

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

Let's say the length of the man's shadow on the building is represented by "x" (in meters). We want to find dx/dt, the rate at which x is changing with respect to time.

Given:
- Distance from the light to the wall (d) = 12 meters
- Speed at which the man is walking toward the building (v) = 1.2 m/s
- Height of the man (h) = 2 meters

Let's set up a similar triangle using the similar properties of triangles and shadows:

Man (height = h)
-----------------
| |
| |
| Shadow | (length = x)
| |
| |
-----------------
Wall (distance = d)

We can see that the two triangles (man-shadow triangle and wall-shadow triangle) are similar. Therefore, we can set up the following proportion:

h / x = (h + d) / (x + dx)

Cross-multiplying, we get:

h * (x + dx) = x * (h + d)

Expanding the equation, we get:

hx + h * dx = hx + xd

We can cancel out the common term hx from both sides:

h * dx = xd

Now, we can solve for dx/dt, by dividing both sides of the equation by dt:

(dx/dt) * h = x * (dh/dt)

Since (dh/dt) is the vertical speed at which the man's height is changing (which is 0 since the man's height is not changing), it can be ignored:

(dx/dt) * h = 0

Now, let's substitute the given values into the equation:

(1.2) * 2 = x * 0

2.4 = 0

Since the equation leads to an impossibility (2.4 = 0), it means that the rate at which the length of the man's shadow on the building is decreasing is 0.

Therefore, when the man is 4 meters from the building, the length of his shadow on the building is not changing or decreasing.