Question

1. a building with a height of 14 m casts a shadow that is 16 m long while a taller building casts a 24 m long shadow. what is the height of the taller building?
2. katie uses a copy machine to enlarge her rectangular design that is 6 in. wide and 8 in. long. the new width is 10 in. what is the new length?

Explanation:

Step1: Set up proportion for building problem

Let \( x \) be the height of the taller building. Since the ratio of height to shadow length is the same for both buildings, we have \(\frac{14}{16}=\frac{x}{24}\).

Step2: Solve for \( x \) in building problem

Cross - multiply: \(16x = 14\times24\). Then \(x=\frac{14\times24}{16}\). Simplify: \(x=\frac{14\times3}{2}=21\) m.

Step1: Set up proportion for Katie's design problem

Let \( y \) be the new length. The ratio of width to length is the same for the original and enlarged design. So \(\frac{6}{8}=\frac{10}{y}\).

Step2: Solve for \( y \) in Katie's design problem

Cross - multiply: \(6y = 8\times10\). Then \(y=\frac{8\times10}{6}=\frac{40}{3}\approx13.33\) in.

Answer:

1. The height of the taller building is \(21\) m.
2. The new length is \(\frac{40}{3}\approx13.33\) in.