A 12-pack box of soda cans has dimensions 4.875 inches by 5.125 inches by 15.5 inches. A single can of soda is approximately a cylinder with a diameter of 2.5 inches and a height of 4.875 inches. What percent of the full box volume is not taken up by the cans, and thus ‘wasted space’? Show your calculations, including units as appropriate.
The Correct Answer and Explanation is:
Calculation of Wasted Space in the Soda Box
Step 1: Compute the Volume of the Box
The box has dimensions:
- Length = 15.5 inches
- Width = 5.125 inches
- Height = 4.875 inches
The volume of the box is given by:
Vbox=length×width×heightV_{\text{box}} = \text{length} \times \text{width} \times \text{height}
Vbox=15.5×5.125×4.875V_{\text{box}} = 15.5 \times 5.125 \times 4.875
Vbox=388.83 cubic inchesV_{\text{box}} = 388.83 \text{ cubic inches}
Step 2: Compute the Volume of a Single Soda Can
Each soda can is modeled as a cylinder with:
- Diameter = 2.5 inches → Radius = 2.52=1.25\frac{2.5}{2} = 1.25 inches
- Height = 4.875 inches
The volume of a cylinder is calculated using:
Vcan=πr2hV_{\text{can}} = \pi r^2 h
Vcan=π(1.25)2(4.875)V_{\text{can}} = \pi (1.25)^2 (4.875)
Vcan≈3.1416×7.6172V_{\text{can}} \approx 3.1416 \times 7.6172
Vcan≈23.92 cubic inchesV_{\text{can}} \approx 23.92 \text{ cubic inches}
Step 3: Compute Total Volume of 12 Cans
V12 cans=12×VcanV_{\text{12 cans}} = 12 \times V_{\text{can}}
V12 cans=12×23.92V_{\text{12 cans}} = 12 \times 23.92
V12 cans=287.04 cubic inchesV_{\text{12 cans}} = 287.04 \text{ cubic inches}
Step 4: Compute Wasted Space
Vwasted=Vbox−V12 cansV_{\text{wasted}} = V_{\text{box}} – V_{\text{12 cans}}
Vwasted=388.83−287.04V_{\text{wasted}} = 388.83 – 287.04
Vwasted=101.79 cubic inchesV_{\text{wasted}} = 101.79 \text{ cubic inches}
Step 5: Compute Percentage of Wasted Space
%wasted=(VwastedVbox)×100\%_{\text{wasted}} = \left(\frac{V_{\text{wasted}}}{V_{\text{box}}}\right) \times 100
%wasted=(101.79388.83)×100\%_{\text{wasted}} = \left(\frac{101.79}{388.83}\right) \times 100
%wasted≈26.18%\%_{\text{wasted}} \approx 26.18\%
Explanation
The wasted space in the soda box arises because the cans do not perfectly fill the rectangular container. The shape mismatch between cylindrical cans and the box’s rectangular shape results in empty volume. The calculations confirm that approximately 26.18% of the total box volume is unoccupied by the soda cans. This space accounts for gaps between cans and the unused areas around their curved surfaces. Efficient packaging designs often aim to reduce wasted space by optimizing the arrangement of products. However, in this case, the box remains larger than the total volume of the cans due to structural constraints and storage considerations. The presence of wasted space may contribute to packaging inefficiencies and excess material usage.
