A box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides as in the figure below.

Suppose that the box height is h=3 in. and that it is constructed using 137 in.2 of cardboard (i.e., AB=137). Which values A and B maximize the volume?

Question

hillnickimsancy hillnickimsancy

04-11-2016
Mathematics

contestada

A box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides as in the figure below.

Suppose that the box height is h=3 in. and that it is constructed using 137 in.2 of cardboard (i.e., AB=137). Which values A and B maximize the volume?

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MissPhiladelphia MissPhiladelphia · Answer 1 · 2016-11-09T14:24:02+01:00

We first identify the given values, AB = 137. With this information we can have an equation for any of the two unknown. Let's say A = 137/B.

The volume of the box would be V = LWH, where it becomes V = 3(A-6) (B-6) and we substitute the equation with our first one to make it

V = 3(137/B-6)(B-6) and simplifying it we have, V= 519- 2466/B - 18B

Next we get its derivative so V' = -18 -2466/B^2 = 0 where we have B = √137 or 11.70.

In getting A, we substitute it A = 137/11.70 = 11.70. The values of A and B are the same, √137 or 11.70 inches.