Colton is 1.55 meters tall. At 1 p.m., he measures the length of a tree's shadow to be 44.35 meters. He stands 39 meters away from the tree, so that the tip of his shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.

Question

39 m
44.35 m
 1.55 m
(Diagram is not to scale.)

Answer 1

To find the height of the tree using the information given, we can use the concept of similar triangles.

We have Colton, whose height is 1.55 meters, and the length of his shadow is 44.35 meters. The distance from Colton to the tree is 39 meters.

Let \( h \) be the height of the tree and the length of the tree's shadow, \( s \), can be expressed as the sum of Colton's shadow length and the distance from Colton to the tree: \[ s = 44.35 + 39 = 83.35 \text{ meters.} \]

Now, we can set up a proportion based on the similar triangles formed by the heights and shadows:

\[ \frac{h}{83.35} = \frac{1.55}{44.35}. \]

To find \( h \), we cross-multiply: \[ h \cdot 44.35 = 1.55 \cdot 83.35. \]

Calculating \( 1.55 \cdot 83.35 \): \[ 1.55 \cdot 83.35 \approx 129.2625. \]

Now, replacing that in our equation: \[ h \cdot 44.35 = 129.2625. \]

To solve for \( h \): \[ h = \frac{129.2625}{44.35} \approx 2.91. \]

Thus, the height of the tree, rounded to the nearest hundredth of a meter, is: \[ \boxed{2.91} \text{ meters.} \]