Question

practice:

1. on a sunny day, sara’s shadow is 5 m long while the shadow cast by a tree was 8 m long. if sara is 1.6 m tall, determine the height of the tree.
2. john’s eyes are at a height of 1.7 m. he stands 10 m from a shed and holds a 12 cm pencil 80 cm away by extending his arm. the bottom of the pencil is directly in front of his eyes and the top of the pencil is in the line of sight from his eyes to the top of the building. determine the height of the shed.
3. a 25 cm tall puppy looks into a mirror on the ground and sees its owner’s face which is 1.8 m above the ground. if the puppy is 0.2 m away from the mirror, determine the distance to the owner from the puppy.

the following triangles.

Explanation:

Problem 1:

Step 1: Set up proportion

Let \( h \) be the height of the tree. We use similar triangles, so \(\frac{\text{Height of Sara}}{\text{Length of Sara's shadow}}=\frac{\text{Height of tree}}{\text{Length of tree's shadow}}\), which gives \(\frac{1.6}{5}=\frac{h}{8}\).

Step 2: Solve for \( h \)

Cross - multiply: \( 5h = 1.6\times8 \). Then \( 5h=12.8 \), and \( h=\frac{12.8}{5}=2.56 \) m.

Step 1: Set up proportion

Let \( H \) be the height of the shed. The ratio of the height of the pencil to the distance from John to the pencil should be equal to the ratio of the height of the shed to the distance from John to the shed. The height of the pencil is \( 12\space cm = 0.12\space m \), the distance from John to the pencil is \( 80\space cm=0.8\space m \), and the distance from John to the shed is \( 10\space m \). So \(\frac{0.12}{0.8}=\frac{H - 1.7}{10}\) (we subtract John's eye height from the shed's height).

Step 2: Solve for \( H \)

Cross - multiply: \( 0.8\times(H - 1.7)=0.12\times10 \). Then \( 0.8H-1.36 = 1.2 \). Add \( 1.36 \) to both sides: \( 0.8H=1.2 + 1.36=2.56 \). Divide by \( 0.8 \): \( H=\frac{2.56}{0.8}=3.2+1.7 = 4.9 \) m.

Step 1: Set up proportion

Let \( d \) be the distance from the puppy to the mirror. We use similar triangles. The ratio of the height of the puppy to the distance from the puppy to the mirror is equal to the ratio of the height of the owner's face above the ground to the distance from the owner to the mirror. The height of the puppy is \( 25\space cm = 0.25\space m \), the height of the owner's face above the ground is \( 1.8\space m \), and the distance from the owner to the mirror is \( 0.2\space m \). So \(\frac{0.25}{d}=\frac{1.8}{0.2}\).

Step 2: Solve for \( d \)

Cross - multiply: \( 1.8d=0.25\times0.2 \). Then \( 1.8d = 0.05 \), and \( d=\frac{0.05}{1.8}=\frac{1}{36}\approx0.0278 \) m or \( 2.78\space cm \).

Answer:

The height of the tree is \( 2.56 \) m.

Problem 2: