The Problem

In the late afternoon, a person who is 6 feet tall casts a shadow 15 feet long. If the top of their shadow is at the same point on the ground as the top of the shadow of a 50 foot tree, how far is the man from standing from the base of the tree?

(Source: Adapted from Mathematics Teaching in the Middle School, Sep1995).

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The Solution

We can solve this problem using the ratio relationship between similar triangles. The figure make triangles, and since the smaller triangle shares a vertex with the larger triangle, and they are both right trianlges they are similar. The relationship between the objects can be better understood looking at the picture below:



We can solve for the missing distance by setting up a ratio and solving for the unknown side:

6ft/50ft = 15ft/x

We can cross multiply and develop the following equation:

6x = 750

x= 125 feet

So we can see that the tree's shadow is 125 long.