Question

1. determine the surface area of each of the shipping boxes. complete the last column. note: because you are finding all sides of a rectangular prism, you can use the formula sa = 2lw + 2wh + 2lh. 3. according to your table, which shipping box requires the least material to build? 4. which shipping box requires the most material to build? 5. what do you notice about the effect of dimensions on surface area? 6. what is the cost of the cardboard for one shipping box that requires the least material? 7. what is the total cost of the cardboard for one shipping box that requires the most material? 8. what other considerations might be important to consider when shipping the boxes?

Explanation:

Since the dimensions of the shipping - boxes are not given, we can't calculate the surface areas and costs precisely. But we can still provide a general step - by - step approach for questions 2 - 7 and an answer for question 8.

Step1: Calculate surface area for each box

For a rectangular prism with length $l$, width $w$, and height $h$, use the formula $SA = 2lw+2wh + 2lh$. Once the dimensions of each box are known, substitute the values into the formula to find the surface area of each box.

Step2: Compare surface areas

After calculating the surface areas of all the boxes, compare the values. The box with the smallest surface - area value requires the least material to build, and the box with the largest surface - area value requires the most material to build.

Step3: Calculate cost for least and most material boxes

If the cost per unit area of the cardboard is $C$, then the cost of the cardboard for a box with surface area $SA$ is $Cost=SA\times C$. For the box with the least surface area $SA_{min}$, the cost $Cost_{min}=SA_{min}\times C$. For the box with the most surface area $SA_{max}$, the cost $Cost_{max}=SA_{max}\times C$.

Answer:

1. Without box dimensions, we can't calculate the surface areas. But use the formula $SA = 2lw + 2wh+2lh$ for each box.
2. Once surface areas are calculated, the box with the smallest surface - area value.
3. Once surface areas are calculated, the box with the largest surface - area value.
4. Generally, larger dimensions result in a larger surface area. Also, the shape (how the dimensions are proportioned) can affect the surface area. For example, a cube (where $l = w=h$) has a different surface - area to volume ratio compared to a long, thin rectangular prism with the same volume.
5. Once the surface area of the box with the least material is calculated as $SA_{min}$, and if the cost per unit area of cardboard is $C$, the cost is $SA_{min}\times C$.
6. Once the surface area of the box with the most material is calculated as $SA_{max}$, and if the cost per unit area of cardboard is $C$, the cost is $SA_{max}\times C$.
7. Other considerations might include:

• The volume of the box to ensure it can hold the items to be shipped.
• The strength and durability of the box to protect the contents during transit.
• The weight of the box, as it can affect shipping costs.
• The ease of handling and stacking of the boxes.