The base of an aquarium with a given volume V is made of slate and the sides are made of glass. If slate costs five times as much(per unit area) as glass, find the dimensions of the aquarium that minimize the cost of materials.
The base should be a square. Compared to all other rectangles, a square has the smallest circumference for a given area.
If the length of one side of the base is x, the total volume is
V=x^2*h
h=V/(x^2)
So the cost is 5 times the area of the base + 4 times the area of the sides
5*x*x+4*x*h
= 5*x*x+4*x*(V/(x^2))
= 5*x*x+4*V/x
Now you need to find the minimum of that equation
10*x-4*V/(x^2)=0
10*x=4*V/(x^2)
10*x^3=4*V
x=(4*V/10)^(1/3) References :
November 28th, 2009 at 11:43 pm
its 5x by x
References :
November 29th, 2009 at 12:26 am
The base should be a square. Compared to all other rectangles, a square has the smallest circumference for a given area.
If the length of one side of the base is x, the total volume is
V=x^2*h
h=V/(x^2)
So the cost is 5 times the area of the base + 4 times the area of the sides
5*x*x+4*x*h
= 5*x*x+4*x*(V/(x^2))
= 5*x*x+4*V/x
Now you need to find the minimum of that equation
10*x-4*V/(x^2)=0
10*x=4*V/(x^2)
10*x^3=4*V
x=(4*V/10)^(1/3)
References :