These problems all require the use of the angle properties of a circle together with Pythagoras' theorem. You'll find a thread that deals with these here:
Have a look and see if that's enough to help you out with these. If you post your answers to those you can do, that'll give me a better idea of where you need help.
Cuts have got to leave congruent pieces so four cuts at right angles to the axis and equally spaced would be one way. This would make 5 equal shorter cylinders.
Or you could make the cuts through a diameter of the circular ends and along the axis. These cuts would have to be 45 degrees apart.
And there are several ways that mix these two types of cut. I'll leave you to find them.
Have a look here:
Welcome to the forum.
You must have some sort of input device and viewer to use this forum. So these three; <input> .... <forum> .... <view answer> form such a machine. It's a powerful one too because you can ask your question in a variety of ways and there'll usually be someone who will give you a mathematical answer. Other 'machines' are not so flexible
The second hand is the radius and it sweeps out circles as time passes. So firstly work out the circumference of that circle (2 times pi times radius) then work out how many circuits it will do in that time . One per minute. Don't use 3.14 or other for pi ... just leave the answer a N.pi where N is a number.
MathWA: I don't see the picture either, but, do you really need it for this question?
Welcome to the forum.
You should have a line of useful symbols at the top of each forum page which includes this one: ≥
Part (a). You can re-write this as a + a + ... + a (n of these) + 1 + 2 + ... + n-1.
The sum of n-1 natural numbers is given by n(n-1)/2
So you can put the answer for part (a) equal to 100 and look for possible whole numbered solutions.
In the instruction page for asking for help it says "We are happy to help! But we don't do your homework for you."
I thin that applies to AoPS problems too.
So, if I help, I try to provide hints to the method rather than just posting an answer. If you would like such hints please post again and I'll do so.
Welcome to the forum.
I followed this back to the AoPS page and got a 'lack of the right font' error message.
I agree with ishankhare's interpretation since it actually leads to a problem that can be done.
Some mathematicians use a little m to mean 'the measure of'. It can be used for either a length or an angle measure so I avoid it myself as I think it creates confusion.
Firstly you're told the (angle) measure of the arc RS. If C is the centre of the circle this means that angle RCS is 40 degrees.
There is a Euclidean theorem that the angle at the centre is twice the angle at the edge of the circumference (see http://www.mathisfunforum.com/viewtopic.php?id=17799 posts 6 and 7)
So angle RWS = 20 = angle RTS.
So the problem makes sense if it's asking what these two angles add up to.
Hope that helps,
This problem can be done by using the 'triangle inequalty'. If x, y and z are the sides of any triangle then x < y + z
THis is an outline of what I did:
(a) AD < AB + BD
Write another expression for AD and add them together.
(b) You can get one half of the inequality by using the first part. Write two more expressions like this for BE and CF. Then add them together.
Then AB < BG + AG and similar expressions for BC and AC. Add these together and also use 3BG = 2BG and two similar.
No difference. There's no absolute authority on which maths symbols to use or even how to spell math. So you'll find both are used. It's only the squiggle on the page after all. It's the square root of minus 1 either way. Or negative one. And that could be ee-th er or ahy-th er
Welcome to the forum.
No apology needed. Post away. Your queries may be the only ones I can do, so you'll be making me feel better at least.
Throughout my life I learnt no music nor how to play an instrument. When I retired I decided to build my own electric guitar. Still struggling to play but I did discover some interesting maths in those frets. If you like, I'll explain in easy to follow steps.
I've not heard of that. No, my example isn't even about geometry.
Back when I taught classes, I had a sequence of lessons on 'The Nature of Proof'. It only touched the surface, of course, but I was trying to show why mathematicians want to prove things and give some ideas about how they go about it.
My introduction was this:
Consider the quadratic
Pick a counting number for n and evaluate the formula. Is the result a prime number?
n=1, formula = 43 This is a prime.
n=2, formula = 47 This is a prime.
n=3, formula = 53 This is a prime.
n= 20, formula = 461 This is a prime.
You might like to try a few values for n yourself.
By the time I'd been round the class and everyone had had a go, they were happy to agree that this formula generates primes.
If you've spotted a value of n that disproves this, well done!
If you set it up on a spreadsheet, you can try out a lot of values. You will find it is 'prime heavy', at least for quite a while; I haven't tried all values of n.
And that's the point. Just because a result keeps on happening, doesn't mean it will keep on happening.
What is the first value of n which doesn't give a prime? And the next ? Say this one is 'm'. You'll find that m+1 gives another prime, and then there's another long sequence of primes before the next non prime.
Conclusion. Just because an 'experiment' generates lots of confirming results doesn't prove that result is true.
I've probably told this before (sorry) but in another lesson a pupil taught me something. For years I had done the following:
Get everyone to draw a triangle (any will do); measure the angles and add them up. If you have a class of 30, you'll get lots of answers such as 178, 179, 180, 181, 182. Up until this particular day the classes had always been happy to conclude that the angles of a triangle add up to 180 and the variation is just experimental error chiefly because of the thickness of the datum lines on the protractor. But this one time a pupil came up with a different answer. He said the angles of a triangle always add up to a number that is close to 180. On the basis of the data you cannot fault him. Worth a gold star I think.
Anyway, do tell me about the Euclid conjecture. Thanks,
I used Euclidean geometry and algebra:
Let AXB = ABX = x : XBC = y : and ACB = z
x = y + z (external angle of a triangle XBC is sum of two internal opposites) ..... (1)
x + y - z = 39 (given)
Substitute (1) into the above
y + z + y - z = 39 => 2y = 39 => y = 19.5
ps. Would you like to see an example to show why experimental maths is not always reliable ?
In your final line you have y = 3 or -1. Clearly the negative cannot be a solution to the original.
Incorrect answers can arise when you use squaring as a way to solve an equation. Consider this simple example:
x = 3 => x^2 = 9 => x^2 - 9 = 0 => (x-3)(x+3) = 0 => x = 3 or x = -3
So it is important to consider if an apparent answer actually fits the problem set. -1 doesn't. There is no more significance to that than x = -3 in my example.
Here's another: A rectangle is 2 units longer than it is wide. Its area is 15. What are its measurements?
Let the width be x. Then the length is x + 2 and x(x + 2) = 15 => x^2 + 2x - 15 = 0 => (x + 5)(x - 3) = 0 => x = 3 or -5
So the measurements are 3 and 5, or -5 and -3. Clearly this second 'answer' cannot be right as a rectangle cannot have a negative measurement.
This last example shows again how spurious answers can crop up.
I'll leave you to work out which is a solution and why the other appears.