Questions: A 5-foot tall student cast a 4-foot shadow. If the tree next to her cast a 44-foot shadow, what is the height of the tree? (A) 35 1/2 ft (B) 45 ft (C) 51 1/2 ft (D) 55 ft

A 5-foot tall student cast a 4-foot shadow. If the tree next to her cast a 44-foot shadow, what is the height of the tree?
(A) 35 1/2 ft
(B) 45 ft
(C) 51 1/2 ft
(D) 55 ft

Transcript text: A 5-foot tall student cast a 4-foot shadow. If the tree next to her cast a 44-foot shadow, what is the height of the tree? (A) $35 \frac{1}{2} \mathrm{ft}$ (B) 45 ft (C) $51 \frac{1}{2} \mathrm{ft}$ (D) 55 ft

Solution

Solution Steps

Step 1: Understand the problem

The problem involves similar triangles. The student and the tree both cast shadows due to the sun's rays, which are parallel. This means the triangles formed by the student and her shadow, and the tree and its shadow, are similar. Therefore, the ratios of their corresponding sides are equal.

Step 2: Set up the proportion

Let $ h $ be the height of the tree. The height of the student is 5 feet, and her shadow is 4 feet. The tree's shadow is 44 feet. Using the property of similar triangles: \[ \frac{\text{Height of student}}{\text{Shadow of student}} = \frac{\text{Height of tree}}{\text{Shadow of tree}} \] Substitute the known values: \[ \frac{5}{4} = \frac{h}{44} \]

Step 3: Solve for $ h $

Cross-multiply to solve for $ h $: \[ 5 \times 44 = 4 \times h \] \[ 220 = 4h \] Divide both sides by 4: \[ h = \frac{220}{4} = 55 \]

The height of the tree is $ 55 $ feet.