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?

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UpStudy AI · Accepted Answer

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**.
Jacob will cast a shadow of approximately 5.09 feet. The lighthouse will cast a shadow of approximately 42.67 feet.

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