Glen is building a rectangular frame for a flower box with 30 m of lumber. Use toothpicks to investigate the greatest area that Glen can enclose.

a) Let each toothpick represent 1 m of lumber. Construct different rectangles to represent the flower box's area. Record the dimension and the area in each case.

b) How many different rectangles are possible?

c) Which shape would you choose for the flower box? Give reasons for your choice.

A rectangular children's play area is to have an area of 36 m^2. The play area will be enclosed by edging bricks which will form the perimeter of the play area.

a) On grid paper, sketch all the rectangles with whole-number dimensions and an area of 36 m^2.

b) Record the dimensions and the perimeter in each case.

c) Which shape would be the most economical for the garden? Why?

A mirror is to have an area of 4 m^2. What should the dimensions of the mirror be to minimize the amount of framing required to go around the outside?

What is the maximum area of a rectangular horse paddock that can be enclosed with 160 m of fencing in each case?

The yard is enclosed on all four sides.

What is the maximum area of a rectangular horse paddock that can be enclosed with 160 m of fencing in each case?

The yard is enclosed on three sides.

A rectangular parking lot is to have an area of 800 m^2. The parking lot is surrounded by a chain-link fence.

a) What are the dimensions of the parking lot that can be enclosed most economically? Round the dimensions to the nearest tenth of a metre.

b) Give reasons why the parking lot might not be designed in the most economical shape that you determined in part a).

Cookies are to be packaged in a square-based prism box with a capacity of 950 cm^3. Use a table like the one shown, or the spreadsheet you created in Section 9.3 to determine the dimensions of the box that requires the least amount of material. Round the dimensions of to the nearest tenth of a metre.

Sea salt is packaged in a plastic-coated square-based prism box with a capacity of 802.125 mL.

a) Determine the dimensions of the box that requires the minimum amount of material. Round the dimensions to the nearest tenth of a centimetre.

b) Explain why these dimensions might not be the ones the manufacturer chooses.

A 2-L box of instant mashed potatoes is a square-based prism and is to be made from the minimum amount of cardboard. Determine the minimum amount of cardboard required, to the nearest square centimetre.

Use a table like the one shown, or the spreadsheet you created in Section 9.4, to investigate the dimensions of the square-based prism box with maximum volume that can be made from 3 m^2 of cardboard. Round the dimensions to the nearest hundredth of a metre.

What are the dimensions of the square-based prism box with maximum volume that can be made from 3000 cm^2 of cardboard? Round the dimensions to the nearest tenth of a centimetre.

Suppose the cardboard in question 10 is a rectangular sheet that measures 30 cm by 100 cm. Explain why it may not be possible to make the shape you determined.

Use a table like the one shown, or the spreadsheet you created in Section 9.5 to investigate the dimensions of the cylinder with maximum volume that can be formed using 620 cm^2 of cardboard. Round your answer to the nearest hundredth.

A manufacturer is trying to choose the best package for white rice. A square-based prism and a cylinder require the same amount of cardboard to make. Which shape should the manufacturer choose? Give reasons for your answer.

a) Use a table like the one shown, or the spreadsheet you created in Section 9.6, to determine the minimum amount of aluminum required to make a pop can with a capacity of 450 mL. Round your answers to the nearest hundredth

b) What assumptions did you make in your solution?