A small jewelry box with square of base is to have a volume of 125 cu.cm. Find its dimensions to require the least amount of material.
The Volume of a box with a square base x by x cm and height h cm is
The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.
The surface area of the box described is
We need A as a function of x alone, so we'll use the fact that
which gives us ,so the area becomes:
We want to minimize A, so
when
The only critical number is
The second derivative test verifies that A has a minimum at this critical number:
which is positive at
The box should have base cm by cm and height 3.1498 cm.
Comments
Leave a comment