Question

practice 2
a board game manufacturer wraps its game boxes in plastic. its most popular game comes in a box thats 8 cm tall and uses 1,408 square centimeters of plastic wrap. the company sells a travel version of the game in a box thats a dilation of the original box. the travel version uses 198 square centimeters of plastic wrap.
how tall is the travel versions box?
type your answer in the box.
_____ cm

Explanation:

Step1: Recall the relationship between surface area and scale factor for similar solids.

For similar solids, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions (like height). Let the scale factor be \( k \), the surface area of the original box be \( S_1 = 1408 \, \text{cm}^2 \), the surface area of the travel version be \( S_2 = 198 \, \text{cm}^2 \), the height of the original box be \( h_1 = 8 \, \text{cm} \), and the height of the travel version be \( h_2 \). The formula for the ratio of surface areas is \( \frac{S_2}{S_1}=k^2 \).

Step2: Calculate the scale factor.

First, find \( k^2=\frac{198}{1408} \). Simplify this fraction: \( \frac{198\div22}{1408\div22}=\frac{9}{64} \). Then, take the square root of both sides to find \( k \). Since \( k \) is a positive scale factor (it's a dilation, so lengths are scaled by a positive number), \( k = \sqrt{\frac{9}{64}}=\frac{3}{8} \).

Step3: Find the height of the travel version.

The relationship between the heights is \( h_2 = k\times h_1 \). Substitute \( k = \frac{3}{8} \) and \( h_1 = 8 \, \text{cm} \) into the formula: \( h_2=\frac{3}{8}\times8 \). The 8 in the numerator and denominator cancels out, leaving \( h_2 = 3 \, \text{cm} \).

Answer:

3