Let's solve the given problem step-by-step:
Abel is 6 feet tall and casts an 8-foot shadow. A lamppost next to him casts a 13.5-foot shadow. The two triangles formed (one by Abel and his shadow and the other by the lamppost and its shadow) are similar because the angle to the sun is the same for both.
Since the triangles are similar, the corresponding sides are proportional. This means we can set up a proportion involving the heights and the shadows of Abel and the lamppost:
[tex]\[
\frac{\text{Abel's height}}{\text{Abel's shadow}} = \frac{\text{Lamppost's height}}{\text{Lamppost's shadow}}
\][/tex]
Given the measurements:
- Abel's height = 6 feet
- Abel's shadow = 8 feet
- Lamppost's shadow = 13.5 feet
- Let [tex]\( h \)[/tex] be the height of the lamppost
So we have:
[tex]\[
\frac{6}{8} = \frac{h}{13.5}
\][/tex]
This can be rearranged to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{6 \cdot 13.5}{8}
\][/tex]
Additionally, another valid equation can be derived by cross-multiplying the original proportion:
[tex]\[
6 \cdot 13.5 = 8 \cdot h
\][/tex]
or
[tex]\[
6h = 8 \cdot 13.5
\][/tex]
To summarize, the two valid equations that can be used to find the height [tex]\( h \)[/tex] of the lamppost are:
1. [tex]\(\frac{6}{8} = \frac{h}{13.5}\)[/tex]
2. [tex]\(6h = 8 \cdot 13.5\)[/tex]
