To find: The function that models the surface area of a open box with square base whose volume is .
The function that models the surface area of given box is .
Let the dimension of square base be x and height h
Surface area of a open box with square base is,
Volume of the box is,
Substitute 12 for in above equation.
Divide both sides of above equation by .
Summarize all the information as shown in the table below.
|In Words||In Algebra|
|Length of side of square base.|
|Height of the box|
Use the information in the table and model the function.
Thus, the function that models the surface area of given box is .
To find: The dimensions of the box which minimize the material used to make the box.
The length of square base of box is approximately and height is .
The function as calculated in part (a) is,
Sketch the graph of above function as shown below.
Observe from the graph shown in Figure (1) that it attains minimum value at
Length of square base is for minimum surface area.
Height of box as calculated in terms of length of base in part (a) is,
Substitute for x in above equation and solve for h.
Thus, the length of square base of box is approximately and height is .
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