Shadows

By Page Bird

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 feet tree, how far is the man from standing from the base of the tree?

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

I approached this problem using GSP. The ratio of the shadow to the height is 2.5. So I created a point on the x-axis and then using dilation I made a point on the y-axis, which represents the shadow length.

After this step, I decided to trace the point where the two lines intersect. This gave the path of the linear relationship between the height and the shadow length.

The coordinates of the point of intersection are given below. Also, the ratio between the coordinates is 2.5.

Based on this, the relationship is that y or the shadow is 2.5 times the height of the object.

Algebraically, y = 2.5x

The question asks about the distance of the man from the tree when their shadows end at the same point of 50 meters.

The answer is 35 meters because 50 —15 is 35. However, I continued on with this problem by looking at the line below which goes through the point (0, 35). This is the position of the man because if his shadow is 15 meters and ends at 50 meters then he must be standing 35 meters from the tree.

The equation of this line is y = 2.5x + 35. This line was constructed in the same way that y = 2.5x was constructed.

Click here to look at the GSP file of this sketch.

Another way to look at this problem is by using similar triangles.

We know from the picture that 15 feet plus the man’s distance from the tree is equal to 50 feet. Since 15 + 35 = 50, we know that the man’s distance is 35 feet from the tree.