To solve this problem, it is important to understand the relationship between the surface area and the volume of similar objects. In the case of the two chests which are similar rectangular prisms, we can make use of the proportionality properties of similar figures in geometry.
1. Surface Area Ratio Determination:
- The amount of paint used is directly proportional to the surface area of the chests.
- The larger chest uses 16 pints of paint.
- The smaller chest uses 9 pints of paint.
Hence, the ratio of their surface areas is:
[tex]\[
\text{Surface area ratio} = \frac{\text{Surface area of larger chest}}{\text{Surface area of smaller chest}} = \frac{16}{9}
\][/tex]
2. Linear Dimensions Ratio:
- For similar shapes, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions.
- Let the ratio of the linear dimensions be [tex]\( r \)[/tex].
Since the surface area ratio is given by [tex]\( r^2 \)[/tex], we have:
[tex]\[
r^2 = \frac{16}{9}
\][/tex]
Solving for [tex]\( r \)[/tex], we take the square root of both sides:
[tex]\[
r = \sqrt{\frac{16}{9}} = \frac{4}{3}
\][/tex]
3. Volume Ratio:
- The volumes of similar objects are proportional to the cube of the ratio of their corresponding linear dimensions.
Thus, the volume ratio of the larger chest to the smaller chest will be:
[tex]\[
\text{Volume ratio} = r^3 = \left( \frac{4}{3} \right)^3
\][/tex]
Calculating the cube:
[tex]\[
\left( \frac{4}{3} \right)^3 = \frac{4^3}{3^3} = \frac{64}{27}
\][/tex]
Therefore, the ratio of the volume of the larger chest compared to the smaller chest is:
[tex]\[
\boxed{\frac{64}{27}}
\][/tex]
