Question

1. janiya is 50 inches tall. she wants to estimate the height of a palm tree near her school. she measures her shadow and the palm tree’s shadow. she then makes a sketch as shown below.

image of a palm tree with shadow 30 ft, janiya with height 50 in and shadow 56.6 in (or similar, based on sketch), and multiple-choice options: 30 ft, 27.5 ft, 13.5 ft, 3.6 ft

which of these is the best estimate of the height of the palm tree?
hint: 1 ft = 12 in.

Explanation:

Step1: Convert Iniyan's height to feet

Iniyan's height is 60 inches. Since \( 1\ \text{ft} = 12\ \text{in} \), we calculate \( \frac{60}{12} = 5\ \text{ft} \).

Step2: Set up proportion for similar triangles

Let \( h \) be the height of the palm tree. The ratio of height to shadow length should be equal for Iniyan and the palm tree. So, \( \frac{5\ \text{ft}}{3\ \text{ft}}=\frac{h}{18\ \text{ft}} \) (assuming Iniyan's shadow is 3 ft, as per the sketch's 3 ft label).

Step3: Solve for \( h \)

Cross - multiply: \( 3h = 5\times18 \). Then \( 3h = 90 \), and \( h=\frac{90}{3}=30\ \text{ft} \)? Wait, no, maybe Iniyan's shadow is 3 ft (from the 3 ft label near her). Wait, let's re - check. Wait, the palm tree's shadow is 18 ft (from the 18 ft label). Iniyan's height: 60 inches = 5 ft, her shadow is 3 ft (as per the diagram: 3 ft next to her). So the proportion is \( \frac{\text{height of Iniyan}}{\text{shadow of Iniyan}}=\frac{\text{height of palm tree}}{\text{shadow of palm tree}} \). So \( \frac{5}{3}=\frac{h}{18} \). Cross - multiply: \( 3h = 5\times18 = 90 \), so \( h = 30 \) ft? Wait, but the options have 30 ft as an option. Wait, maybe I misread the shadow lengths. Wait, the palm tree's shadow is 18 ft (the 18 ft label), Iniyan's height is 60 inches = 5 ft, her shadow is 3 ft (the 3 ft label next to her). So the proportion is correct. So solving \( \frac{5}{3}=\frac{h}{18} \), we get \( h=\frac{5\times18}{3}=30 \) ft.

Answer:

30 ft (the option with 30 ft)