Question

autumn is 1.25 meters tall. at 10 a.m., she measures the length of a trees shadow to be 22.85 meters. she stands 18.3 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.

Explanation:

Step1: Determine Autumn's shadow length

Let \( x \) be Autumn's shadow length. The tree's shadow length is \( 22.85 \) meters, and Autumn is \( 18.3 \) meters from the tree. So, \( x + 18.3=22.85 \). Solving for \( x \), we get \( x = 22.85 - 18.3=4.55 \) meters.

Step2: Set up proportion for similar triangles

Let \( h \) be the tree's height. Since the triangles formed by Autumn and her shadow, and the tree and its shadow are similar, we have the proportion \( \frac{1.25}{4.55}=\frac{h}{22.85} \).

Step3: Solve for \( h \)

Cross - multiply: \( 4.55h = 1.25\times22.85 \). Calculate \( 1.25\times22.85 = 28.5625 \). Then \( h=\frac{28.5625}{4.55}\approx6.28 \) meters.

Answer:

\( 6.28 \) meters