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What dimensions will result in a box with the largest possible volume if an open rectangular box with square base is to be made from #48 ft^2# of material?

Answer 1

The base will be # 4 xx 4# and the height will be #2# (all numbers in feet)

Let #b# be the length and the width of the base (length and width are the same since the base is square).

Let #h# be the height of the box.

The surface area of the box is #base + height xx perimeter# #= b^2 + 4bh = 48# From which we can determine: #h = (48 - b^2)/(4b)#

The Volume of the box: # V(b) = b^2h = b^2 * ((48 - b^2)/(4b))# # = 12b - (3b^3)/4#

The Volume is a maximum when #( d V(b))/ (db) = 0#

#(d V(b))/(db) = 12 - (3 b^2)/4 = 0#

#b = +-4# (only #+4# is not extraneous)

Plugging this back into the formula #h = (48 - b^2)/(4b)# we get #h = 2#

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Answer 2

Ezra Adams

To maximize the volume of the box, the dimensions should be: length = width = height = 4 feet.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.