A closed box with a square base has a volume of 250 cubic meters. The material for the top and bottom of the box cost $2 per square meter, and the material for the sides cost $1 per square meter. Can the box be constructed for less then $3000?
A closed box with a square base has a volume of 250 cubic meters. The material for the top and bottom of the box cost $2 per square meter, and the material for the sides cost $1 per square meter. Can the box be constructed for less then $3000?
Let the side of the base be of length x and the height y.
The volume is then x^2y = 250.
Top + Bottom = x^2 + x^2 = 2x^2 square meters, at 2 dollars each, this gives a top+bottom cost of 4x^2.
Sides' area = 4xy, at one dollar per square meter, this gives a cost of 4xy dollars
Total Cost = $\displaystyle 4xy + 4x^2$
Restriction: $\displaystyle x^2 y= 250$
Restriction: Total Cost $\displaystyle \leq 3000$
Total cost = $\displaystyle 4x \frac{250}{x^2} + 4x^2 = \frac{1000}{x} + 4x^2$
Optimal cost:
$\displaystyle C'(x) = -1000/x^2 + 8x = 0$
$\displaystyle 8x^3 - 1000 = 0$
$\displaystyle x = 5$
$\displaystyle C(5) = 200 + 100 = 300$
C(5) is a minimum because $\displaystyle C"(x) = \frac{2000}{x^3} + 8 > 0 $ at x=5.
The minimum cost is $300. So then your answer would be no, you cannot build this box with less than 300 dollars, but you can build it with exactly 300 dollars.