9. Two brothers are outside on a hot day. Timothy, who is 5 ft 6 in tall, casts a shadow that is 8 ft in length. If his brother, Jacob is 3 ft 6 in tall, how long of a shadow will Jacob cast? 10. A local lighthouse is 160 feet tall. During a visit to the lighthouse Xavier, who is 6 feet tall, cast a shadow that was 1.6 feet long. How long, in feet, was the shadow that the lighthouse cast?

To solve both problems, we can use the concept of similar triangles, which states that the ratios of corresponding sides are equal. ### Problem 9: Timothy's height and shadow length: - Height of Timothy = 5 ft 6 in = 5.5 ft (since 6 in = 0.5 ft) - Shadow of Timothy = 8 ft Let \( S \) be the length of Jacob's shadow. Jacob's height is: - Height of Jacob = 3 ft 6 in = 3.5 ft Using the ratio of heights to shadows: \[ \frac{\text{Height of Timothy}}{\text{Shadow of Timothy}} = \frac{\text{Height of Jacob}}{S} \] Substituting the values: \[ \frac{5.5}{8} = \frac{3.5}{S} \] Cross-multiplying gives: \[ 5.5S = 3.5 \times 8 \] Calculating the right side: \[ 5.5S = 28 \] Now, solving for \( S \): \[ S = \frac{28}{5.5} \approx 5.09 \text{ ft} \] So, Jacob will cast a shadow of approximately **5.09 feet**. ### Problem 10: Xavier's height and shadow length: - Height of Xavier = 6 ft - Shadow of Xavier = 1.6 ft Let \( L \) be the length of the lighthouse's shadow. The height of the lighthouse is: - Height of lighthouse = 160 ft Using the ratio of heights to shadows: \[ \frac{\text{Height of Xavier}}{\text{Shadow of Xavier}} = \frac{\text{Height of lighthouse}}{L} \] Substituting the values: \[ \frac{6}{1.6} = \frac{160}{L} \] Cross-multiplying gives: \[ 6L = 160 \times 1.6 \] Calculating the right side: \[ 6L = 256 \] Now, solving for \( L \): \[ L = \frac{256}{6} \approx 42.67 \text{ ft} \] So, the lighthouse will cast a shadow of approximately **42.67 feet**.