United States Service has contracted you to design a closed box with a square base that has a volume of 5000 cubic inches. Find a function for a surface area of the box.
height of box = y
width of box = x
length of box = x
Surface area is:
Area of bottom + area of top + area of all 4 sides.
As the base is square, the areas of the top and bottom are both x^2.
The area of each side is xy.
So the total surface area is 2x^2 + 4xy = 2x (x + 2y).
Now we can work out what the side y is in terms of x, because we know that the volume is 5000.
So we know that x times x times y = 5000, or that y = 5000/(x^2).
So we can plug this expression for y into the one for surface area:
Surface area = 2x (x + 2 (5000/(x^2))
A reasonable question to ask next is to find the value of x which makes the surface area a minimum - but only if you've done calculus.