An open top box with a square base of length x and height y

An open top box with a square base of length x and height y is to be constructed so that its volume is V=2660cm^3. If the cost of the material for the base of the box is $0.45 per square centimeter and the cost of the sides is $0.78 per square centimeter, express the cost C of the box in terms of the length of the base x.

Solution

Volume of the box = length * width * height
2660=x*x*y
2660=x^2y
2660/x^2=y...(i)

Area of base= x*x=x^2
cost per unit area for base = $0.45 per cm^2
so cost for base = 0.45x^2...(ii)

Area of vertical faces=4(Area of one vertical face)
{4 times because there are 4 vertical faces having same dimension}
Area of vertical faces=4(xy)
Area of vertical faces=4(x(2660/x^2)) {using (i) }
Area of vertical faces=4*2660/x
Area of vertical faces=10640/x
cost per unit area for sides = $0.78 per cm^2
so total cost for sides = 0.78*(10640/x)=8299.2/x...(iii)

Now total cost is given by adding (ii) and (iii)

so required answer is
C(x)= 0.45x^2 + (8299.2/x)
where C(x) represents cost in terms of x