A prism has height h cm and a square base of side x cm. The Surface area,
A cm2 , of the prism is given by A = 2x2 + 4xh. Find the value of h when A =
192 and x=6.
h = _______ cm
Answers
Answer:
The Volume of a box with a square base x by x cm and height h cm is V=x2h
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 A=x2+4xh
We need A as a function of x alone, so we'll use the fact that
V=x2h=32,000 cm^3
which gives us h=32,000x2, so the area becomes:
A=x2+4x(32,000x2)=x2+128,000x
We want to minimize A, so
A'=2x−128,000x2=0 when 2x3−128,000x2=0
Which occurs when x3−64,000=0 or x=40
The only critical number is x=40 cm.
The second derivative test verifies that A has a minimum at this critical number:
A''=2+256,000x3 which is positive at x=40.
The box should have base 40 cm by 40 cm and height 20 cm.
(use h=32