FOCUS ON CONCEPTS Volume and Surface Area Name______________________ 1. Consider a box shaped as a rectangular prism. The volume of the box is 24 cubic feet. What are the possible dimensions (length x width x height) of the box if the dimensions are whole numbers? What is the surface area of each "possible" box (assume the box has no top)? What are the dimensions of the box with the smallest surface area? You may find it useful to organize your work in a table such as the one below. Box # | Length | Width | Height | Surface Area 2. You plan to construct an open cardboard box. You have 30 square feet of cardboard. What are the possible dimensions (length x width x height) of the box if the dimensions are whole numbers? What is the volume of each "possible" box? What are the dimensions of the box with the greatest volume? You may find it useful to organize your work in a table such as the one below. Box # | Length | Width | Height | Volume 3. Suppose you want to construct a rectangular prism (i.e. a box enclosed on all sides) with the 30 square feet of cardboard. What are the possible dimensions (length x width x height) of the prism if the dimensions are whole numbers?
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You plan to construct an open cardboard box. You have 30 square feet of cardboard. What are the possible dimensions (length x width x height) of the box if the dimensions are whole numbers? What is the volume of each "possible" box? What are the dimensions of the box with the greatest volume? You may find it useful to organize your work in a table such as the one below. Box # Length Width Height Volume
Sri K.
Complete the table that follows for four different rectangular prisms; each of which has a volume of 216 cm³ (to make the calculations easier, you should choose whole-number values for the dimensions): Length (cm) | Width (cm) | Depth (cm) | Volume (cm³) | Surface Area (cm²) | V:SA Ratio (cm) ------------|------------|------------|--------------|--------------------|---------------- Prism 1 | 6 | 6 | 6 | 36 | 6:1 Prism 2 | 8 | 3 | 9 | 46 | 9:2 Prism 3 | 9 | 4 | 6 | 58 | 6:3 Prism 4 | 12 | 3 | 6 | 72 | 6:4 Which of the four prisms from above (Question 6) uses the container material most efficiently? Least efficiently? Explain. Why would a manufacturer be concerned about the ratio of volume to surface area? Why are cereal boxes not shaped to give the greatest ratio of volume to surface area? 10. Design and construct your own package for a product of your choice (you do not need to submit the packaging, just describe it). Draw a net for your package and specify the dimensions, surface area, amount, and type of packaging material used, and volume of the package. Explain why your design is efficient and why it would be a good design for a manufacturer to use.
Jeremiah M.
A rectangular open-top box is to be constructed out of a 9 -inch by 12 -inch sheet of thin cardboard by cutting $x$ -inch squares out of each corner and bending the sides up as shown in Figures 39 and 40 in Example $6 .$ What size squares to two decimal places should be cut out to produce a box with a volume of 72 cubic inches? Give the dimensions to two decimal places of all possible boxes with the given volume.
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