We have a block (a*b*c) with volume of 1 m^3. There are 2 questions:

  1. write surface area of block as function of a and b

  2. find a and b in a way so that surface will be as big as possible

I had luck solving similar problems in the past using derivatives, like surface of garden, etc. But with this one i have no luck. If someone could help me out it would be great. I have an exam in like 4 hours but it doesn't matter if you answer later because it is bugging me, i really want to know the answer. I looked at similar problem with Cylinder but there its easier because you can express surface and volume with only 2 variables, and then if your volume is given at the start, you can easily delete 1 variable and then solve the problem easily.

ps: this problem is taken from this years exam btw


There is no "biggest surface" box of volume $1$.

Make the base be say $100$ metres by $100$ metres. True, the box will not be very tall, the height will be $\frac{1}{10000}$ metres, not very useful for storing things. The surface area is then $\gt 20000$ square metres.

And if that is not big enough, we can get bigger surface area. Let the base be an $m\times m$ square, where $m$ is the distance to the Moon.

Remark: For smallest surface area, we can show that the optimal box has all sides equal to $1$.


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