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The sun causes a shadow that extends 24 feet from the side of a house that is 16 feet tall. Nearby is a 4-foot-fall fence, which also has a shadow. If the sun's rays hit the house and the fence at the same angle, what is the lenght of the shadow cast by the shed.
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Let's say:
- x as the length of the shadow cast by the shed.
- 24 as the length of the shadow cast by the house.
- 16 as the height of the house.
- 4 as the height of the fence.
[tex]\[ \mathsf{ \text{Given: } H = 16 \text{ feet}, \quad S_H = 24 \text{ feet}, \quad F = 4 \text{ feet} } \] [/tex]
[tex]\[ \mathsf{ \text{Using the similarity of triangles:} \quad \frac{H}{S_H} = \frac{F}{S_F} } \][/tex]
Solve for:
[tex]\[ \mathsf{ \text{ } S_F: \quad \frac{16}{24} = \frac{4}{S_F} \Rightarrow S_F = \frac{4 \cdot 24}{16} = 6 \text{ feet} } \][/tex]
[tex]\[ \boxed{\mathsf{\textcolor{brown}{S_F = 6 \text{ feet}}}} \][/tex]
The idea of similar triangles, in which the ratio of the corresponding sides of similar triangles is equal, can be used to calculate this. The height of the fence and the length of its shadow are identical in this instance, as is the ratio of the house's height to that of its shadow.
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