A man is walking away from a lamppost with a light source h = 6 m above the ground. The man is m = 1.5 m tall. How long is the man's shadow when he is d = 11 m from the lamppost? (Hint: Use similar triangles.)

To find the length of the man's shadow, we can use the concept of similar triangles. Let's consider the two triangles formed by the man, his shadow, and the lamppost.

In the first triangle, we have the man's height (1.5 m) as the vertical side, and the length of his shadow (let's call it s) as the horizontal side.

In the second triangle, we have the height of the lamppost (6 m) as the vertical side, and the distance between the man and the lamppost (11 m) as the horizontal side.

Since the two triangles are similar, their corresponding sides are in proportion. Therefore, we can write the following equation:

man's height / length of shadow = height of lamppost / distance between man and lamppost

Plugging in the given values, we can solve for the length of the man's shadow:

1.5 m / s = 6 m / 11 m

Cross-multiplying:

1.5 m * 11 m = 6 m * s

16.5 m = 6 m * s

Dividing both sides by 6 m:

s = 16.5 m / 6 m

s ≈ 2.75 m

Therefore, when the man is 11 m from the lamppost, his shadow is approximately 2.75 m long.

To find the length of the man's shadow, we can use similar triangles.

Let's denote the length of the man's shadow as x.

Since we have a triangle formed by the lamppost, the man, and his shadow, we can set up a proportion based on the similarity of the triangles:

(Length of shadow) / (Distance from man to lamppost) = (Length of man) / (Distance from man to ground)

Using the given values, we can substitute:

x / d = m / h

Plugging in the values, we get:

x / 11 = 1.5 / 6

To isolate x, we can cross-multiply:

x = (11 * 1.5) / 6

Simplifying, we get:

x = 2.75 m

Therefore, the length of the man's shadow when he is 11 m from the lamppost is 2.75 m.

Let the man's shadow length be x.

Draw the figure showing how the shadow is cast.

Similar triangles should tell you that
6/(11 + x) = 1.5/x

Solve for x

6x = 33/2 + 3x/2
9x/2 = 33/2

x = 33/9 = 11/3 = 3.667 m