So we know we have a cylindrical container and there is no top. There is a bottom and we know that it has a radius of our will say the height is age and we know that it holds a volume of cubic centimeters of liquid. So we know that the volume is the area of the bottom times height, and that equals So we have a relationship between R and H and let's solve for H dividing both sides.
We have divided by pi r squared. That's how big H is because we want the variable to We only want one variable. We either wanted in terms of age we want in terms of our and it was so easy to solve for H. So go and we'll go in terms of, uh, have a variable in terms of our now we wanna find how big should r and h b so that we minimize the amount of material.
And so we want to minimize the surface area and the area of Theseus er fis is we have a bottom on the bottom is a circle which is pyre square, and then we have a cylinder and I usually think of If I chopped this right here and I laid that out flat, that this would be a certain height and then all the way around would be the circumference in the circumference is that two pi times are so we know the area of this is have the length, which is two pi r times the height.
But our problem is we don't wanna have this in terms of again both r and H. So we want the area as a function of just the radius, just the radius, which, when you put it in the calculator will actually use X.
But so we have pi r squared plus two pi r times h and let's do a substitution for H. We know that this is what H is equal Thio in terms of our so we're going to substitute in place of H and that they'll chop that right there. That age goes here we want to substitute in place of age that over pi r squared and we can see what ends up happening.
Those pies cancel and that our cancels and this just becomes an are down here. So our area formula in terms of the variable are is pi r squared plus, and then two times is over our. So there's our equation. And when you get to calculus, you'll actually be able to slob this problem without, uh, without using a calculator and you'll be able to do some calculus on it.
But for us, we're gonna put this into our graph for so we will go to wise up one and let's see if I can get this screen to move back over. There we go. My screen won't move up. And so we are going to go on our calculator and let's get that back on.
We're gonna go on our calculator and we're going thio, change the happy wise of one be this equation and I'm gonna put right in here what the window is so I don't have to move My screen. I put this in is why someone using this is an X squared and using this as an ex. And so I had my equation typed into their at high times back squared, plus divided by acts.
And then I made a table of values. And I found that this was a nice window. If I on my calculator, if I went from 0 to 10 and if I went up to prompt and I scaled by once and I went up to you didn't really have to go up that high. But I could see that it started off when you plug one in place The backs It was, I think, , And then it kept going down.
It kept going down, and then it was slowly increasing back up. And what I want to do is figure out what that low is. And so got that rap on my calculator. And then I left bound and right bound, and I found what that value was for that minimum, and that minimum value came out to be within our value off 5.
And then I plugged that value back in here divided by pi r squared and that gave me Ah, height off also 5. So those are the dimensions of the cylinder. We want the radius and the height to be the same about 5. They don't ask it, but it came out to be So this had to be done graphically right now, in order to see what that low value waas. I'm slightly lost on how to approach the problem. I'm thinking Liles Liles 13 1 1 silver badge 3 3 bronze badges. Add a comment.
Active Oldest Votes. Calculus is easier How do I solve for something in Volume, such as h , knowing only ? You should be able to continue from there. Rory Daulton Rory Daulton Why is it multiplied by 2? You should verify this. NoName NoName 2, 1 1 gold badge 11 11 silver badges 32 32 bronze badges. Well, I did say to verify Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.
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