Question

Imelda held a 6-inch long pencil at arm's length so that it matched the height of a tree 100 feet away. If the pencil was 24 inches from her eyes, about how tall was the tree?

Answer 1

25 feet

Explanation:

1/2 foot Pencil is 2 feet away

use similar triangles

1/2 ft is to 2 feet as 'x' is to 100 ft

1/2 / 2 = x / 100

100( 1/2 /2) = 25 feet

Answer 2

The pencil and Imelda's arm together make a right triangle, similar to the triangle made by the tree with her arm plus the 100 feet distance. Those triangles are represented below

Where H represents the height of tree. Before we start calculating, first let's convert all the measures to the same unit. The relation between inches and feet is given below

`1ft=12in\Rightarrow\begin{cases}6in=0.5ft \\ 24in=2ft\end{cases}`

The ratio between corresponding sides on similar triangles is the same, therefore, we have the following equation

`(0.5)/(2)=(H)/(100+2)`

Solving for H, we have

`\begin{gathered} (0.5)/(2)=(H)/(100+2) \\ (0.5)/(2)=(H)/(102) \\ H=102\cdot(0.5)/(2) \\ H=25.5 \end{gathered}`

The height of the tree is 25.5 ft.