A tree casts a shadow 36 ft long. At the same time, the shadow cast by a vertical 5 - ft pole is 4 ft long. Find the height of the tree The tree's height is 37.5 ft .

To determine the height of the tree, we can use the concept of similar triangles. When the sun casts shadows, the ratios of the heights of objects to the lengths of their shadows remain consistent. Here's a step-by-step explanation: ### Given: - **Height of the pole (H₁):** 5 feet - **Shadow of the pole (S₁):** 4 feet - **Shadow of the tree (S₂):** 36 feet - **Height of the tree (H₂):** ? ### Step 1: Understand the Relationship Since both the pole and the tree are vertical and their shadows are cast under the same sunlight, the triangles formed by the pole and its shadow and the tree and its shadow are similar. This means the ratios of their corresponding sides are equal. ### Step 2: Set Up the Proportion \[ \frac{H₁}{S₁} = \frac{H₂}{S₂} \] Plugging in the known values: \[ \frac{5 \text{ ft}}{4 \text{ ft}} = \frac{H₂}{36 \text{ ft}} \] ### Step 3: Solve for \( H₂ \) Cross-multiply to solve for \( H₂ \): \[ 5 \text{ ft} \times 36 \text{ ft} = 4 \text{ ft} \times H₂ \] \[ 180 = 4H₂ \] \[ H₂ = \frac{180}{4} = 45 \text{ ft} \] ### Conclusion: There seems to be a discrepancy between the calculated height and the provided answer of 37.5 feet. Let's double-check the calculations. **Re-evaluating:** If we assume the correct answer is 37.5 feet, let's see what proportion would lead to that: Given: \[ H₂ = 37.5 \text{ ft}, \quad S₂ = 36 \text{ ft} \] \[ \frac{5}{4} = \frac{37.5}{36} \quad \text{? \] Calculating the right side: \[ \frac{37.5}{36} \approx 1.0417 \] Calculating the left side: \[ \frac{5}{4} = 1.25 \] Since \( 1.25 \neq 1.0417 \), the correct height based on the proportion is **45 feet**, not 37.5 feet. ### Final Answer: **The height of the tree is 45 feet.**