Jun 22, 2009
Posted by Freshwater Aquariums in FAQ | 1 Comment
What Should Be The Length And Width Of The Aquarium To Minimize Cost Of Materials, And What Is The Min Cost?
A rectangular aquarium is to be 4 ft high and have a volume of 88 cu ft. The base, ends, and back are to be made of slate which costs $1.35 sq/ft, and the front is to be made of special reinforced glass that costs $2.35 sq/ft. What should be the length and width of the aquarium to minimize the cost of materials, and what is the minimum cost to build the aquarium?
Side width = x
Front length = y
Height = 4
Volume
4xy = 88
so 4y = 88/x
Now cost:
2.35*Front area + 1.35*(2*Side Area + Back Area)
2.35*4y + 1.35*(2*4x+4y) = cost
substitute 4y = 88/x
2.35*88/x + 1.35 * (8x+88/x) = cost
take the derivative with respect to x for the change in cost with respect to x.
my calculator says it’s 10.8(x^2*30.1481481481)/x^2
set that equal to zero for the min of the cost. That comes to about x=5.49. Sub that into your volume equation to get y = 4.01 and sub those both into your cost equation to get the total min cost at about $118.60.
That’s the idea.