Question

A mirror is placed 35 feet from the base of a waterfall by a hiker. The hiker walks backwards until they are 7.5 feet from the mirror. Determine how tall the waterfall is if the hiker is 6 feet tall.
45 ft
38 ft
28 ft
24 ft

Answer 1

To determine the height of the waterfall, we can use the concept of similar triangles.

1. Let's outline the scenario:
- The hiker is 6 feet tall.
- The distance from the hiker to the mirror is 7.5 feet.
- The distance from the base of the waterfall to the mirror is 35 feet.

2. The setup creates two similar triangles:
- Triangle 1 (the hiker's triangle): The height of the hiker is 6 feet, and the distance from the hiker to the mirror is 7.5 feet.
- Triangle 2 (the waterfall's triangle): The height we want to find, which we'll call \( h \), and the distance from the waterfall to the mirror is 35 feet.

3. Set up the proportion based on the similarity of the triangles:

\[
\frac{\text{Height of Hiker}}{\text{Distance from Hiker to Mirror}} = \frac{\text{Height of Waterfall}}{\text{Distance from Waterfall to Mirror}}
\]

Plugging in the known values:

\[
\frac{6 \text{ ft}}{7.5 \text{ ft}} = \frac{h}{35 \text{ ft}}
\]

4. Cross-multiply to solve for \( h \):

\[
6 \cdot 35 = 7.5 \cdot h
\]

\[
210 = 7.5h
\]

5. Divide both sides by 7.5 to isolate \( h \):

\[
h = \frac{210}{7.5}
\]

Simplifying:

\[
h = 28 \text{ ft}
\]

Therefore, the height of the waterfall is **28 feet**.