The tree in the illustration casts a shadow $c = 6$ feet long, while a person $a = 6$ feet tall casts a shadow $b = 4$ feet long.
Find the height of the tree.
A) $\frac { 3 } { 12 }$
B) 15
C) $\frac { 1 } { 9 }$
D) 9
E) none of these

Question

The tree in the illustration casts a shadow $c = 6$  feet long, while a person $a = 6$  feet tall casts a shadow $b = 4$  feet long. 
Find the height of the tree. 
A) $\frac { 3 } { 12 }$ 
B) 15 
C) $\frac { 1 } { 9 }$ 
D) 9 
E) none of these

Quizplus · Accepted Answer

We can use similar triangles to solve this problem. Let's call the height of the tree H. Then, using similar triangles ACE and ABF, we have:

$\frac{H}{6}=\frac{AC}{AB}$

And using similar triangles ADE and ABF, we have:

$\frac{H+6}{6}=\frac{AD}{AB}$

Simplifying both equations, we get:

$AC=\frac{c}{b}a=9$

$AD=\frac{a}{b}c=9$

Substituting these values back into our original equations, we get:

$\frac{H}{6}=\frac{9}{AB}$

$\frac{H+6}{6}=\frac{9}{AB}$

Solving for AB in both equations, we get:

$AB=\frac{9}{\frac{H}{6}}=\frac{54}{H}$

$AB=\frac{9}{\frac{H+6}{6}}=\frac{54}{H+6}$

Setting these two expressions equal to each other, we get:

$\frac{54}{H}=\frac{54}{H+6}$

Cross-multiplying and simplifying, we get:

$H=9$

Therefore, the height of the tree is 9 feet.

The Tree in the Illustration Casts a Shadow c=6c = 6c=6

The Tree in the Illustration Casts a Shadow $c = 6$