SOLUTION: Here is an application problem from a chapter on radicals. Surface area of a cube. The formula A=6V^2/3 gives the surface area of a cube in terms of its volume V. What is the vo

Algebra ->  Radicals -> SOLUTION: Here is an application problem from a chapter on radicals. Surface area of a cube. The formula A=6V^2/3 gives the surface area of a cube in terms of its volume V. What is the vo      Log On


   

Question 277174: Here is an application problem from a chapter on radicals.
Surface area of a cube. The formula A=6V^2/3 gives the surface area of a cube in terms of its volume V. What is the volume of a cube with surface area 12 square feet?
Thanks.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
A=6V%5E%282%2F3%29
You're given A so:
12+=+6V%5E%282%2F3%29
We're solving for V so we need to get rid of the 6 in front and the exponent of 2/3. We can eliminate the 6 by dividing both sides by 6:
2+=+V%5E%282%2F3%29
Now how do we get rid of the exponent? Well, to be perfectly correct, we are not getting rid of the exponent we are changing it to a 1 (since V%5E1+=+V. So how do we change the exponent to a 1?

One way is to raise each side to the 3/2 power. (If you don't see why this works yet, you will shortly.)
%282%29%5E%283%2F2%29+=+%28V%5E%282%2F3%29%29%5E%283%2F2%29
On the right side, the rule is to multiply the exponents. 2/3 and 3/2 are reciprocals of each other. And what do you get when you multiply reciprocals? Answer 1! This is why I raised both sides to the 3/2 power:
  • I knew that raising a power to a power would result in multiplying exponents
  • I knew that I wanted an exponent of 1
  • I knew that multiplying reciprocals always results in 1
  • 3/2 is the reciprocal of 2/3

%282%29%5E%283%2F2%29+=+V
Now we only have to simplify the left side. If you are not yet comfortable with fractional exponents, it can be helpful to look at the exponent in factored form. 3%2F2+=+3%2A%281%2F2%29 The factor of three in the exponent says we'll be cubing something and the factor of 1/2 in the exponent says there will be a square root. And, since multiplication is Commutative, we can do these things in any order we choose. Since 2 is not a prefect square, doing the square root first has little appeal. So I'll start by cubing 2. 2 cubed is 8 and then we apply the square root giving us sqrt%288%29. So:
V+=+2%5E%283%2F2%29+=+sqrt%288%29
The only thing left is to simplify the square root. There is a perfect square factor in 8: