19. MODELING REAL LIFE A saguaro cactus casts a shadow that is 30 feet long. At the same time, a person standing nearby who is 5 feet 10 inches tall casts a shadow that is 50 inches long. How tall is the cactus? Example 3

To determine the height of the saguaro cactus, we can use the concept of similar triangles. The proportion of the height to the shadow length remains consistent for both the cactus and the person. ### Given: - **Cactus Shadow:** 30 feet - **Person's Height:** 5 feet 10 inches - **Person's Shadow:** 50 inches ### Step 1: Convert All Measurements to the Same Unit It's easier to work with consistent units. Let's convert everything to **inches**. 1. **Cactus Shadow:** \( 30 \text{ feet} \times 12 \text{ inches/foot} = 360 \text{ inches} \) 2. **Person's Height:** \( 5 \text{ feet} \times 12 \text{ inches/foot} + 10 \text{ inches} = 60 + 10 = 70 \text{ inches} \) 3. **Person's Shadow:** Already in inches: 50 inches ### Step 2: Set Up the Proportion The ratio of height to shadow length is the same for both the cactus and the person. \[ \frac{\text{Cactus Height}}{\text{Cactus Shadow}} = \frac{\text{Person's Height}}{\text{Person's Shadow}} \] \[ \frac{h}{360} = \frac{70}{50} \] ### Step 3: Solve for \( h \) (Cactus Height) \[ h = 360 \times \frac{70}{50} \] \[ h = 360 \times \frac{7}{5} \] \[ h = 360 \times 1.4 = 504 \text{ inches} \] ### Step 4: Convert Back to Feet \[ 504 \text{ inches} \div 12 \text{ inches/foot} = 42 \text{ feet} \] ### **Conclusion:** The saguaro cactus is **42 feet** tall.