Question

1. a local lighthouse is 173.75 feet tall. during a visit to the lighthouse, tammy, who is 5 feet tall, cast a shadow that was 1.8 feet long. how long, in feet, was the shadow that the lighthouse cast? 96.52 feet

Explanation:

Step1: Set up proportion

Let \( x \) be the length of the lighthouse's shadow. The ratio of height to shadow length should be equal for Tammy and the lighthouse. So, \(\frac{5}{1.8}=\frac{173.75}{x}\).

Step2: Cross - multiply

Cross - multiplying gives \( 5x = 1.8\times173.75 \).

Step3: Calculate right - hand side

First, calculate \( 1.8\times173.75 = 312.75 \).

Step4: Solve for \( x \)

Then, \( x=\frac{312.75}{5}=62.55 \). Wait, there was a miscalculation earlier. Let's recalculate \( 1.8\times173.75 \): \( 173.75\times1.8=(170 + 3.75)\times1.8=170\times1.8+3.75\times1.8 = 306+6.75 = 312.75 \). Then \( x=\frac{312.75}{5}=62.55 \)? Wait, no, wait. Wait, the proportion is height over shadow. Tammy's height is 5, shadow is 1.8. Lighthouse height is 173.75, shadow is \( x \). So \(\frac{5}{1.8}=\frac{173.75}{x}\), cross - multiply: \( 5x=1.8\times173.75 \). \( 173.75\times1.8 = 173.75\times\frac{9}{5}=\frac{173.75\times9}{5}=173.75\times1.8 = 312.75 \). Then \( x=\frac{312.75}{5}=62.55 \)? Wait, but maybe I made a mistake. Wait, 5 divided by 1.8 is approximately 2.777..., and 173.75 divided by 62.55 is approximately 2.777..., which matches. Wait, the original answer was 96.52, which was wrong. Let's do it again.

Wait, maybe the proportion is shadow over height? No, the ratio of height to shadow should be the same because the angle of the sun is the same. So height1/shadow1 = height2/shadow2. So Tammy's height (5) / Tammy's shadow (1.8)=Lighthouse height (173.75)/Lighthouse shadow (x). So \( 5x = 1.8\times173.75 \). \( 1.8\times173.75 = 312.75 \). Then \( x = 312.75\div5 = 62.55 \). Wait, but let's check with another approach. The ratio of Tammy's height to her shadow is 5/1.8. So the lighthouse's shadow is lighthouse height divided by (5/1.8)=lighthouse height(1.8/5)=173.750.36 = 62.55. Yes, that's correct. So the correct length of the lighthouse's shadow is 62.55 feet.

Wait, maybe the user's initial calculation was wrong. Let's go through the steps again:

1. Set up the proportion: Since the triangles formed by the objects and their shadows are similar (same sun angle), the ratios of height to shadow length are equal. So \(\frac{\text{Tammy's height}}{\text{Tammy's shadow}}=\frac{\text{Lighthouse's height}}{\text{Lighthouse's shadow}}\)
2. Let \( x \) be the lighthouse's shadow length. Then \(\frac{5}{1.8}=\frac{173.75}{x}\)
3. Cross - multiply: \( 5x = 1.8\times173.75 \)
4. Calculate \( 1.8\times173.75 \):

• \( 173.75\times1.8=(170 + 3.75)\times1.8=170\times1.8+3.75\times1.8 \)
• \( 170\times1.8 = 306 \), \( 3.75\times1.8 = 6.75 \)
• So \( 306+6.75 = 312.75 \)

1. Solve for \( x \): \( x=\frac{312.75}{5}=62.55 \)

Answer:

62.55