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Problem 61 Hard Difficulty

Find a formula for the described function and state its domain.

An open rectangular box with volume 2 m$ ^3 $ has a square base. Express the surface area of the box as a function of the length of a side of the base.

Answer

$a>0$

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Video Transcript

Okay, so here we have this open top box and we know it's volume, and our goal is to find the surface area as a function of a side link. So we know that the base is a square, so we could label the base with side length A and A and we don't know the height of the box, so we can just call it H. Now we do know the volume is too. So we know volume is length times, width, times, height. So that would be a times a times age, a square times age is too. So perhaps will end up using that relationship as we go. Our goal is to find the surface area and because there's no top, we have only five services. We have the base, which is a times A, and we have four of these sides which are eight times h. So the surface area, which I'm going to call capital all college A s since we're already using an A for something else surface area is s is going to be the some of these areas. So a squared for the area of the base, plus four times a times age for the four sides. Now we need this to be a function of just the side, not the side and the height. So here's where that other equation comes into play. Let's take that other equation we got from volume and isolate H. We get H equals two over a squared and we can substitute that into the equation we're working with to get rid of the H. So now we have s equals a squared plus four, a times two over a squared. Now that's gonna need a little bit of simplifying. So here's that formula again, and we want to simplify it. So we're going to go ahead and multiply the four a by the two over a squared and that gives us eight over a. That should be a plus. Okay, now, this would be perfectly fine. However, if you want to keep simplifying and get a common denominator, you could multiply the first term by a over a. We have a cubed over a plus eight over a and that gives us a cubed plus eight over a. I don't feel strongly that that's necessary. I would be happy to leave it like this, but these are equivalent. All right, What about the domain? So what could a be a represented the length of aside? And so it has to be positive. Other than that, we don't know of any restrictions on it.