# Thread: Optimization and Induction Problems

1. ## Optimization and Induction Problems

Hey guys I'm stuck on two problems and dont really know what to do.

1.A rectangular box with top has base three times its width. Find the dimensions of such a box of maximum volume that can be constructed from 200 square inches of material. (Optimization)
2.Prove by induction: ((i^3)-i)=(n((n^2)-1)(n+2))/4

Any help would be greatly appreciated.

2. ## Re: Optimization and Induction Problems

Originally Posted by Boogiewoogie34
Hey guys I'm stuck on two problems and dont really know what to do.

1.A rectangular box with top has base three times its width. Find the dimensions of such a box of maximum volume that can be constructed from 200 square inches of material. (Optimization)
2.Prove by induction: ((i^3)-i)=(n((n^2)-1)(n+2))/4

Any help would be greatly appreciated.
Please take enough care enough to copy the problems correctly. Neither of these makes any sense. In the first problem you say "has base three times its width". The base has area measured in square units. It cannot be "twice its width" since width is a linear measure. Did you mean the length of the base is three times its width? In the second problem, the left side of the equation, the left side depends on i only while the right side depends on n only. They cannot be equal. Was there a summation on the left?

If the first problem does indeed say that the length of the base is three times its width, then, taking w for the width, the length is 3w and, taking h for the height, the volume is given by $V= 3w^2h$ and the total surface are is $6w^2+ 2wh+ 6wh= 200$. You can solve the surface area equation for h in terms of w, replace h by that in the volume equation, to have an equation in w only. Differentiate with respect to w and set the derivative equal to 0.

If the second problem is really $\sum{i= 1}^n i^3- i= n(n^2- 1)(n+ 2)/4$, to use induction, the first step is to show that it is true for the first case, n= 1.
Then the sum is just 1- 1= 0 while the right side is 1((1- 1)(1+ 2)/4= 0.

Now show that "if this is true for n= k, it is also true for n= k+1." That is, If $\sum_{i=1}^k i^3- i= k(k^2- 1)(k+ 2)/4$ then $\sum_{i=1}^{k+1} i^3- i= (k+1)((k+1)^2- 1)((k+1)+ 2)/4$. Notice that $\sum_{i=1}^{k+1} i^3- i= \sum_{i= 1}^k i^k- i+ ((k+1)^2- (k+1)$.

3. ## Re: Optimization and Induction Problems

Thanks for the help with the optimization problem. I think the problem was worded wrong and I was really confused what to do with it. For the induction problem, you're right it is a summation problem and I understand how to set it up I just don't know how to do the algebra to get to the final k+1 formula