.
Let x be the size of the squared base edge and h be the height.
Then the volume is x^2*h = 100 cubic inches
and the lateral surface area is 4*xh square inches.
The total cost is C(x,y) = 5x^2 + 2*4*xh = 5x^2 + 8xh.
So we need to minimize C(x,y) = 5x^2 + 8xh under restriction x^2*h = 100.
We then express h = and substitute it into the expression for C(x,y).
We then get
C(x,y) = + = + ,
and we need to minimize this function.
Find the derivative and equate it to zero
10x - = 0.
From this equation, find x
10*x^3 = 800
x^3 = = 80.
x = = 4.309 inches.
Then h = = = 5.386 inches.
ANSWER. The base size is 4.309 inches; the height is 5.386 inches.
Solved.
------------------
If you want to see many other similar solved problems, look into the lesson
- Calculus optimization problems
in this site.