Question

Suppose the length and width of the box in exercise 12 double. Does the surface area S double? Explain

Answer 1

No, it does not.

In general, surface area is the sum of the areas of each face of a prism.  In a rectangular prism, we have length, width, and height.  

The area of the bottom of the prism and the top of the prism is the same, and can be represented as l*w.  This means that together, it is 2lw.

The area of the left and right faces of the prism is the same, and can be represented as l*h.  This means that together, it is 2lh.

The area of the front and back faces of the prism is the same, and can be represented as w*h.  This means that together, it is 2wh.

This gives us the equation

SA = 2lw+2lh+2wh

If we double the length and width, making them each 2l, we then have:

SA = 2(2l*2w) + 2(2l*h) + 2(2w*h)
      = 2(4lw) + 4lh + 4wh
      = 8lw + 4lh + 4wh

If the surface area had been doubled, we would expect 
4lw + 4lh + 4wh

No, it is not doubled.