Question

gabriella is 1.25 meters tall. at 3 p.m., she measures the length of a trees shadow to be 15.45 meters. she stands 10.2 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: Find the length of Gabriella's shadow

The length of the tree's shadow is \(15.45\) meters and Gabriella stands \(10.2\) meters away from the tree. So the length of Gabriella's shadow \(x=15.45 - 10.2=5.25\) meters.

Step2: Set up a proportion

Let \(h\) be the height of the tree. We can set up a proportion using the fact that the ratio of height to shadow length is the same for Gabriella and the tree.
For Gabriella, her height is \(1.25\) meters and her shadow length is \(5.25\) meters. For the tree, its height is \(h\) and its shadow length is \(15.45\) meters.
The proportion is \(\frac{1.25}{5.25}=\frac{h}{15.45}\).

Step3: Solve the proportion for \(h\)

Cross - multiply: \(5.25h=1.25\times15.45\).
First, calculate \(1.25\times15.45 = 19.3125\).
Then \(h=\frac{19.3125}{5.25}\).
\(h = 3.68\) (rounded to the nearest hundredth).

Answer:

\(3.68\) meters