Some airlines have restrictions on the size of items of luggage that passengers

Question

Some airlines have restrictions on the size of items of luggage that passengers are allowed to take with them. Suppose that one has a rule that the sum of the length, width and height of any piece of luggage must be less than or equal to 156 cm. A passenger wants to take a box of the maximum allowable volume. If the length and width are to be equal, what should the dimensions be?
length = width =
height =
In this case, what is the volume?
volume =
(for each, include units)

If the length is be twice the width, what should the dimensions be?
length =
width =
height =
In this case, what is the volume?
volume =
(for each, include units)

Explanation / Answer

given l+ w +h = 156

now volume = lwh

first case if l and w are same

volume = l^2*h

for max volume

dv/dt =0

differentiating on both sides

0 = l^2 + 2hl

l = 2h

and w =2h

so substituing we get

l + w +h = 156

=> 5h=156

=> h= 31.2 cm

l = 62.4 cm

w = 62.4cm

volume = 62.4 * 62.4 * 31.2

=121485.312 cm^3

if l =2w

then V = 2w * w *h

=> V = 2(w^2)*h

now for max volume differentiatiing both sides we get

0 = 2w^2 + 4wh

=> w =2h

so l =4h

l + w + h =156

=> h = 156/7 =22.28 cm

=> w = 2h = 44.57cm

=> l = 4h =89.14cm

volume = 88520.604 cm^3

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