Question

1. lance the alien is 5 feet tall. his shadow is 8 feet long. at the same time of day, a trees shadow is 32 feet long. what is the height of the tree?

a. 20 feet
b. 24 feet
c. 51 feet
d. 29 feet

1. the shadow cast by a one - foot ruler is 8 feet long. at the same time, the shadow cast by a pine tree is 24 feet long. what is the height, in feet, of the pine tree?

a. 3 feet
b. 16 feet
c. 36 feet
d. 192 feet

1. triangle pqr is similar to triangle wxy.

which proportion describes the relationship between corresponding sides of the triangles?
a. $\frac{qr}{xy}=\frac{6}{3}$
b. $\frac{pq}{wx}=\frac{2}{4}$
c. $\frac{qr}{wx}=\frac{3}{4}$
d. $\frac{pq}{xy}=\frac{2}{6}$

Explanation:

Step1: Set up proportion for 16

Since the ratios of height to shadow - length are equal for similar - shaped objects at the same time of day, we set up the proportion $\frac{\text{height of alien}}{\text{shadow of alien}}=\frac{\text{height of tree}}{\text{shadow of tree}}$. Let the height of the tree be $x$. So, $\frac{5}{8}=\frac{x}{32}$.

Step2: Solve for $x$ in 16

Cross - multiply: $8x = 5\times32$. Then $8x=160$, and $x = 20$.

Step3: Set up proportion for 17

Let the height of the pine tree be $y$. Using the same principle of equal ratios of height to shadow - length, we have $\frac{\text{height of ruler}}{\text{shadow of ruler}}=\frac{\text{height of pine tree}}{\text{shadow of pine tree}}$, so $\frac{1}{8}=\frac{y}{24}$.

Step4: Solve for $y$ in 17

Cross - multiply: $8y=1\times24$, then $y = 3$.

Step5: Analyze similar triangles in 18

For similar triangles $\triangle PQR$ and $\triangle WXY$, the ratios of corresponding sides are equal. The ratio of side $PQ$ in $\triangle PQR$ to side $WX$ in $\triangle WXY$ is $\frac{PQ}{WX}=\frac{2}{4}$.

Answer:

1. A. 20 feet
2. A. 3 feet
3. B. $\frac{PQ}{WX}=\frac{2}{4}$