Question

kayla is 1.85 meters tall. at 12 noon, she measures the length of a trees shadow to be 28.15 meters. she stands 23.1 meters away from the tree, so that the tip of her shadow meets the tip of the trees shadow. find the height of the tree to the nearest hundredth of a meter.
(diagram is not to scale.)

Explanation:

Step1: Find the length of Kayla's shadow

The length of Kayla's shadow is \(28.15 - 23.1=5.05\) meters.

Step2: Set up a proportion

Let \(h\) be the height of the tree. Since the triangles formed by Kayla and her shadow and the tree and its shadow are similar, we have the proportion \(\frac{h}{28.15}=\frac{1.85}{5.05}\).

Step3: Solve for \(h\)

Cross - multiply: \(h=\frac{1.85\times28.15}{5.05}\).
First, calculate \(1.85\times28.15 = 1.85\times(28 + 0.15)=1.85\times28+1.85\times0.15=51.8+0.2775 = 52.0775\).
Then, \(h=\frac{52.0775}{5.05}\approx9.85\).

Answer:

\(9.85\)