Question

do the math use math for design improvement
in order to minimize costs for the manufacture, processed food is often packaged in containers that allow the maximum storage volume but use the minimal amount of material. the best design solution for a cereal box can be chosen by calculating the box dimensions that best meet the criteria of a maximum volume while using the least amount of cardboard.
the table below shows some calculations that engineers made while creating a cereal box. your task is to find the cereal box size that has the maximum volume and the least surface area, or that uses the least amount of cardboard to make the box. the design criteria have been set are as follows:

• the volume must be between 3,400 cm³ and 3,425 cm³.
• the height must be between 25 cm and 27 cm.
• the length must be between 18 cm and 20 cm.
• the width must be between 6 cm and 8 cm.

containers with a large surface area allow the manufacturer to better advertise their product.

1. for each set of box dimensions, determine the volume - to - surface - area ratio. round the results to two decimal places. choose the box that best meets the criteria and explain your reasoning.

Explanation:

Step1: Recall volume - to - surface - area ratio formula

The formula for the volume - to - surface - area ratio is $\frac{V}{SA}$, where $V$ is the volume and $SA$ is the surface area.

Step2: Calculate the ratio for each row

For the first row: $\frac{3417}{1630}\approx2.096\approx2.10$
For the second row: $\frac{3437}{1642}\approx2.093\approx2.09$
For the third row: $\frac{3424}{1626}\approx2.106\approx2.11$
For the fourth row: $\frac{3409}{1609}\approx2.118\approx2.12$
For the fifth row: $\frac{3392}{1592}\approx2.131\approx2.13$
For the sixth row: $\frac{3451}{1611}\approx2.142\approx2.14$
For the seventh row: $\frac{3434}{1595}\approx2.153\approx2.15$
For the eighth row: $\frac{3416}{1578}\approx2.165\approx2.17$
For the ninth row: $\frac{3397}{1562}\approx2.175\approx2.18$

Step3: Select the box that meets the criteria

The volume must be between $3400\ cm^{3}$ and $3425\ cm^{3}$, the height between $25\ cm$ and $27\ cm$, the length between $18\ cm$ and $20\ cm$, and the width between $6\ cm$ and $8\ cm$.
The fourth row has a volume of $3409\ cm^{3}$, height of $26.0\ cm$, length of $19.0\ cm$, and width of $6.9\ cm$ which meets all the criteria.

Answer:

The box with height $26.0\ cm$, length $19.0\ cm$, width $6.9\ cm$, volume $3409\ cm^{3}$, surface - area $1609\ cm^{2}$ and volume - to - surface - area ratio of approximately $2.12$ best meets the criteria. This is because its volume is within the required range ($3400 - 3425\ cm^{3}$), and its dimensions (height, length, width) are also within the specified ranges.