Expert: Alon Mandes - 8/29/2008

Question
a closed box has a base twice as long as it is wide. if its volume is 100 cubic inches, express its surface area as a function of the width, x, of its base

Answer
A box is compounded from 6 plains :
1. 2 bases (up & down), we will call them B1 & B2
2. 2 side plains (left & right) we will call them L & R
3. 2 plains (front & back) we will call them F & B
The surface area is the sum of the area of all the plains :
B1+B2+L+R+F+B. Ok, now all we have to do is find the area of each
as a function of x. Let's get to work :

Let's start with the base : the width is x & the length is 2x.
So the area B1=x*2x=2x^2, which is the same as B2.

Let's see what happens with the sides :
Let's take the right side R : the area is x*h, where h is the height of the box. But, we need h as function of x !! so we use the volume
information : 100=x*2x*h --> h=50/x^2. So, R=x*50/x^2=50/x. The same as L also.
Now, the front & back plains : F=2x*h=2x*50/x^2=100/x. The same also as B.

Let's sum now :  
surface area= B1   + B2   + L    + R    + F     + B
           = 2x^2 + 2x^2 + 50/x + 50/x + 100/x + 100/x
           = 4x^2+300/x


Alon.