Question

to estimate the height of a tree, tia and felix walk away from the tree until the angle of sight with the top and bottom of the tree is a right angle. let h represent the height of a person’s eyes and d represent the distance away from the tree. answer parts a to c below. a. if the height of tia’s eyes is 1.6 m and her distance away from the tree is 2.5 m, what is the height of the tree? the height of the tree is about \boxed{} (type an integer or decimal rounded to the nearest hundredth as needed)

Explanation:

Step1: Identify similar triangles

We have two similar right triangles: one formed by Tia's eyes to the tree's base and the segment of the tree below the line of sight to the top, and the other formed by Tia's eyes to the tree's top and the segment of the tree above the line of sight to the base. For similar triangles, the ratios of corresponding sides are equal:
$\frac{\text{Height of tree above eye level}}{d} = \frac{d}{\text{Height of tree below eye level}}$
Here, the height of the tree below eye level is equal to Tia's eye height, $1.6$ m, and $d=2.5$ m. Let $x$ be the height of the tree above eye level.

Step2: Solve for height above eye level

Substitute the known values into the ratio:
$x = \frac{d^2}{\text{Eye height}} = \frac{2.5^2}{1.6}$
Calculate $2.5^2 = 6.25$, then $\frac{6.25}{1.6} = 3.90625$ m

Step3: Calculate total tree height

Add the eye height to the height above eye level:
$h = x + 1.6 = 3.90625 + 1.6$

Answer:

$5.51$ m