Question

Manufacturers often package products in a way that uses the least amount of material. One measure of the effi ciency of a package is the ratio of its surface area to its volume. The smaller the ratio, the more effi cient the packaging. a. Write an expression for the efficiency ratio \(\frac{S}{V}\) of a cylindrical package. b. Find the efficiency ratio for each cylindrical can listed in the table. $$ \begin{array}{|l|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & \text { Soup } & \text { Coffee } & \text { Paint } \\ \hline \text { Height, } \boldsymbol{x} & 10.2 \mathrm{~cm} & 15.9 \mathrm{~cm} & 19.4 \mathrm{~cm} \\ \hline \text { Radius, } \boldsymbol{r} & 3.4 \mathrm{~cm} & 7.8 \mathrm{~cm} & 8.4 \mathrm{~cm} \\ \hline \end{array} $$ c. Rank the three cans in part (b) according to efficiency. Explain.

Short answer

The efficiency ratio of the Soup, Coffee, and Paint packages are 2.57, 1.83, and 1.60 respectively. Ranked from most to least efficient, the sequence is: Paint, Coffee, Soup.

Step-by-step solution

Derivation of the efficiency ratio expression

The surface area is given by \(S = 2\pi r(r + h)\) and the volume is given by \(V = \pi r^2h\). To find the ratio \(\frac{S}{V}\), just divide the surface area by the volume to give \(\frac{S}{V} = \frac{2\pi r(r + h)}{\pi r^2h} = \frac{2(r + h)}{rh}\).

Calculation of the efficiency ratio of each package

Substitute the given height and radius values for each package into the expression for the efficiency ratio. For Soup: \(\frac{2(3.4 + 10.2)}{3.4*10.2} = 2.57\). For Coffee: \(\frac{2(7.8 + 15.9)}{7.8*15.9} = 1.83\). For Paint: \(\frac{2(8.4 + 19.4)}{8.4*19.4} = 1.60\).

Ranking the packages based on their efficiency ratios

The Paint package is the most efficient because it has the smallest efficiency ratio (1.60), followed by the Coffee package (1.83) and finally the Soup package (2.57). The smaller the efficiency ratio, the more efficient the packaging.

Key concepts

Surface Area and Volume

Cylindrical items, like cans, have two important measures: surface area and volume. Surface area is the total area that the surface of a cylinder covers. It includes the two circular ends and the side, or the wrap-around part. You calculate it with the formula: \( S = 2\pi r(r + h) \). Here, \( r \) is the radius of the cylinder's base and \( h \) is its height. The 2 accounts for both circular ends.
Volume, on the other hand, tells us how much space is inside the cylinder. For cylinders, it's calculated as: \( V = \pi r^2h \). This means you find the area of the base and multiply it by how tall the cylinder is.
In packaging, we use these to understand efficiency. A lower surface area to volume ratio means you’re using less material for the same space inside. This is important for cost and sustainability reasons.

Cylindrical Geometry

In geometry, cylinders are defined by their circular bases and straight sides. They can often be visualized as a can or tube. Understanding cylinder properties is key in calculating their dimensions like radius and height, which directly affect their surface area and volume.
In our problem, each package, like soup or paint cans, can be described as a cylinder. Each has a height \( x \) and a radius \( r \). When you know these two measurements, you can determine how much packaging material you need and how it compares to the space inside. Cylindrical geometry helps in modeling these shapes mathematically, making it easier to design efficient packaging solutions.
This understanding plays a significant role in industries where packaging is involved, ensuring products are stored conveniently while maximizing storage efficiency.

Mathematical Modeling in Packaging

Mathematical modeling allows manufacturers to determine the most efficient package design. In packaging, it involves calculating the surface area to volume ratio (or efficiency ratio) of containers.
For cylindrical packages, we derive the efficiency ratio \( \frac{S}{V} = \frac{2(r + h)}{rh} \). This formula gives insight into which package design is better by comparing their efficiency values.
The efficiency ratio helps rank different packaging options, like soup, coffee, and paint cans, by their usage of materials. In the exercise, smaller ratios indicated more efficient designs. Manufacturers can apply these calculations when designing and selecting packaging to lower costs and environmental impact.