dimis
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posted April 20, 2010 09:19 AM
I think the point everyone is trying to pass across, is that you should figure out the formula, and then you will also know why.
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The empty set
When you don't even know where to begin...nevermind. Yeah I have a bit of a learning dissability sometimes. Get a bit frustrated. Thanks to those who at least tried.
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Myt, if you just wanted the formula you should have just asked. We thought you wanted us to answer a homework problem for you.
Your shape is a cone with the top chopped off. If you know the formula of a cone, you can get the answer by (look at Joonas's pic) subtracting the "missing" cone from the whole cone. This is involves finding the volume of two cones and subtracting the missing portion from the whole cone.
No, I didn't want just the formula. I wanted to know the formula and WHY. The only problem is I get a bit frustrated sometimes on the simplest of things, and have made many teachers throw up their hands in frustration. All because of an accident I had what seems a lifetime ago. It was curiosity, NOT a homework problem.
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I'm afraid you'll become even more frustrated, when diving into there. Anyway nowadays standard methods to find such volums(rotation solids etc) is itegration.
Just googled it and found an article. Please see it. Seems helpful.
Something I thought was interesting about calculus is that for several years of math it was mostly a bunch of theory. I don't know how other people learned it, but when I learned math we just learned theory and skipped most of the application sections in the book.
Then in calculus it suddenly became practical again. In geometry we just memorized all those formulas for volume and surface areas. Then in calculus it was like "ooooh, so THAT is where those formulas came from".
No pun intended, but the math came full circle. We kept advancing until we were back where we were before, but with a whole new perspective on it.
I'll probably be in bed before someone answers this, but I have a question.
Can someone explain the difference between RMS (root mean square) and a "slice" in integration? And why would you use one or the other in a given application? They seem very similar, but are not the same. Also, can one be derived or extrapolated from the other?
Keep in mind I've forgotten all my math, so try to keep the answer as basic as possible.
Guys, I kinda need help with an equation (or whatever) I had in a math program in school, spent hours on it and couldn't solve it. I hope some of you can.
If you have a camera that records a race trough a hallway. It was 75 metres long. The camera took 30 pictures/second and each 'step' forward in the movie is 5 images, how fast is the one doing the race? Anwser in m/s.
The movie showed a hallway where the camera image went from one end to the other, the movie was 6 seconds long.
Now I tried everything I can, but couldn't solve it, my math teacher couldn't solve it.
and before anyone asks about those steps, I don't know, it said the exact same in the version on the program (was in swedish). And you don't even see anyone running in the movie.
BTW the final number is without decimals.
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Why can't you save anybody?
Ok, Mytical, I hope this helps you. Forgive me for squishing a lot into each figure but since there's no equation editor here I had to make do to save time.
How to Derive the Formula for Volume of a Cone
Before deriving the formula for a generalized cone, let’s use a specific example. This cone has a height of 10 and a base of 10 (radius 5).
The first thing we’ll do is lay the cone down on its side. This will make things easier. The x axis runs through the center of the cone and the base is situated along the y axis, as shown below (left side - note, there's an erroneous "minus" sign in the volume formula). Note that the "top edge" of the cone can be described using a linear equation, which will be important later.
Using the formula we see that the cone should have a volume of about 262.
Let’s calculate a volume without using a formula, and then generalize that method to back-generate the formula. We will need to use some calculus to do this.
The general approach is as follows. We can approximate the volume of the cone as the sum of a series of cylinders of equal heights, as shown above (right side).
For the moment, these cylinders have finite width. The volume of a cylinder is just the area of the base times the height. So if we add up all the cylinders (see the equation in the figure, where t is the thickness of each cylinder) we would approximate the volume of the cone. You can see that there is a pretty large margin of error. However, if we made the cylinders thinner, and used more of them, the error would shrink. And if we use cylinders of infinitesimally small thickness, and used an infinitely large number of them, the sum of all the cylinder volumes would approach the real volume of the cone.
This is essentially the basis of an integration. The question is: how do we know what the volume of each successive cylinder is? This is actually easy. The area of the base of each cylinder is determined by the “edges” of the cone. In fact, as you can see from the first figure, we can use a simple line equation (y = mx + b, where m is the slope and b is the y-intercept) to determine what the boundary of the cone is, and thus where the base of each cylinder is. Thus we can easily calculate what the area of the base is for each cylinder. For example, a cylinder whose base is at a value of x = 1 would have a radius of 4.5 and a base area of pi*4.5^2.
When we do our integration of an infinite number of cylinders, then, we use equation 1 (figure below).
Let’s look at this equation for a moment. The integrand is simply the area of the base of each cylinder (pi * radius squared) times the thickness of each cylinder (dx). The radius is a function of x determined by the line equation specified above. The limits of integration (0 and 10) tell us to integrate along the entire length of the cone, from the base (x = 0) to the tip (x = 10). If we perform this integration, we get equation 2 (figure below).
That’s good that we get the same value!
So, let’s return to the general problem.
The only difference here is that we replace our specific values with generalized ones. The base has a radius or r and the height is h. Our line equation for the edge of the cone is provided in the figure, as is the generalized formula of integration. If you perform the integration and do some algebra, you’ll find that it reduces to the formula for a cone shown in the first figure.
Determining the Volume of a Truncated Cone (Frustrum):
Ok, now what about a truncated cone? There are two methods to solve this. Lacking any formula, the easiest way is to do what has been mentioned here already (method 1). Specifically, we recognize that a truncated cone is just a really big cone minus a tip, which is a smaller cone. The one trick is figuring out what the height of the large cone is. That is, what is the length of the truncated part? Well, we can do that using a line equation and then extrapolating it to zero, as shown below using your original problem of a glass with base diameter = 8 and top diameter = 6 and height = 12. The figure also gives volume solutions for the two methods (see below).
Since you know what part of the cone is, you can specify a linear equation for the edge of the cone, find out where it crosses the x-axis, and then calculate the height (and volume) of the large cone using the formula derived above. The smaller “tip” cone is also easily calculated. The volume of the red shaded area is the difference between these two values.
However, a second method is using a formula which you provided for a truncated cone (frustrum) (reproduced in the figure above) and you asked where the formula came from. Can you now guess? We basically use the same integration method as for a whole cone, but integrate not to the tip but to where the cone is cut off.
For a generalized conic frustrum:
This is exactly the same integration method used above for a whole cone, except the integration limits are changed. Our line equation looks a little more complicated, but it really isn’t – recall in the first case that the r2 value was zero. You’ll notice that if you put a value of zero in your formula for r2, it reduces to the simplified formula for a whole cone, because that’s what it becomes if your second radius is zero. That’s a nice check.
Homework:
That's right, Mytical, I'm giving you homework. You should be able to derive the volume formula for a pyramid with a square base of side R. Bonus points: derive the volume formula for a pyramid with an equilateral triangle base of length L). And, for bonus bonus points, the volume of a sphere of radius R. It's not necessary to solve the integrals, but you should, if you understand the explanation above, be able to set up the problem and describe the integrals you would need to solve. [those last two involve some tricky geometry, so beware ]
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I'm sick of following my dreams. I'm just going to ask them where they're goin', and hook up with them later. -Mitch Hedberg
I have enough homework from my Finite math at the moment, we are nearing Midterms. So, when I can, will see what I can do about your pyramid. One question though, you want the 'long' version, or the short one? ____________
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I've got another math thing for you, shouldn't be too hard for you guys, took me 20 minutes to solve it, or rather, took me 20 minutes to find out how to solve it, then it was easy. Here we go:
The CHiang family decided to take the train to Tokyo from their old farm, the train was new and shiny and the trip to Tokyo was 57 miles long. They paid a total of 1755 Yuan for their tickets. Grandma and Grandpa got a 50% discount because they were retired, and Ling got 25% discount because she was a student. How much did Ling's parents pay each for their tickets?
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Why can't you save anybody?
Quote:I've got another math thing for you, shouldn't be too hard for you guys, took me 20 minutes to solve it, or rather, took me 20 minutes to find out how to solve it, then it was easy. Here we go:
The CHiang family decided to take the train to Tokyo from their old farm, the train was new and shiny and the trip to Tokyo was 57 miles long. They paid a total of 1755 Yuan for their tickets. Grandma and Grandpa got a 50% discount because they were retired, and Ling got 25% discount because she was a student. How much did Ling's parents pay each for their tickets?
57 miles is irrelevant.
1755 is unequally split.
So both gran'ma and grandpa pays 50%? Ok.
Ling pays 3/4? Ok.
Parents pay full price? Ok.
They are 5 people? Ok. That means we need to get how much each part is.
4+4+3+2+2=15? Ok.
1755/15=117? Ok each part is 117, that means that each of Lings parents will pay 4 timed that.
117*4=468? So each of Lings parents will pay 468.
This thread is pages long: 
posted April 20, 2010 09:19 AM
You should be able to derive the volume formula for a pyramid with a square base of side R. Bonus points: derive the volume formula for a pyramid with an equilateral triangle base of length L). And, for bonus bonus points, the volume of a sphere of radius R. It's not necessary to solve the integrals, but you should, if you understand the explanation above, be able to set up the problem and describe the integrals you would need to solve. [those last two involve some tricky geometry, so beware 
]
