Math Is Fun Forum

It happens to us all. I wasted a large chunk of an exam trying to eliminate a variable and failing.  Afterwards I spoke to a friend who had done it with one simple thing that I'd missed. Ggggrrr!

Bob

I've found time to look at the video.  The measurements are different from the problem I met years ago and the working comes out more easily, so I thought it would be simplest to make a new picture.

If the barn wasn't there then the goat could reach all the grass inside a circle radius 10, so (pi times 10 times 10) in area.

But the barn gets in the way of the rope and there's no grass to eat inside the barn anyway. The orange area is the bit that the goat can reach without any rope snagging problems.  Hence 3/4 x pi x 10 squared.

When the goat is on the line AD produced, the point D acts as a new tether point with new radius 10 - 7.  Once again the goat cannot reach the whole of the new circle; only 1/4 of a circle radius 3.

Similarly when the goat is on the line AB produced the point B cats as a new tether point with new radius 10 - 9.  So the extra area here is 1/4 of a circle radius 1.

I agree with the video answer.

Bob

Ok. So try one here.

Bob

It sounds like you have to find out how many seconds in that year (not a leap year) and divide by 43.

But you also have to consider what variance there might be in making that assessment.  I'd certainly want to round off my answer ... decimal place figures after the point would have no meaning ... but even the whole number answer probably needs rounding further.  So two people could get different answers as a result of different amounts of rounding I suppose.  But all answers should be in the same ball park.  Someone must have known the total in order to get the 'every 43 seconds' claim.

Bob

Yes! I'm assuming it means he ran  a total of T km in 5 days and his mean daily distance was 13.2 km. Work out T.

Without the word 'daily' it could mean all sorts of things.  Maybe that's his hourly rate!  (OK that's unlikely for a runner but math questions need to be worded clearly.)

Maybe the terrain was very rough and the temperature very hot and so the questioner thought it was mean for the organisers to set such a challenge.

Bob

hi locallycompact

Welcome to the forum.

Whoops, it looks like you are right.  I tried this on a dummy account and got no email.  I can do some things as an administrator but not fix that. Sorry.

Do you have another username that you want to resurrect?  I can probably fix that for you. Tell me the old membership name and I'll see what I can do. I will have to be convinced that you're not trying to steal someone else's account of course.

Bob

hi Chenfeng Liu

Welcome to the forum.

Also you may post questions here for help.

Bob

I tried googling that quote again and most hits attribute it to Galileo but a similar one  "I cannot teach anybody anything, I can only make them think" is attributed to Socrates.  I think you're right.  Galileo would have read Socrates and the two quotes are very similar especially when you take account of translation variations.

A modern version is "You can lead a horse to water, but you can't make him drink."

You can also use the method to find the highest common factor.

eg. Find the HCF of 45 and 72

45 =                 3 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3

This time extract all the primes that occur in both.  So 3 x 3

If you create a Venn diagram with all the primes and each circle enclosing those for one of the  numbers then the lowest common multiple (LCM) is the union of the sets and the HCF is the intersection.

Bob

Reduce each demoninator to its prime factors

3 = 3
9 = 3x3
4 = 2x2
16 = 2x2x2x2
10 = 2x5

Write the largest power of each prime and multiply them

(2x2x2x2) x (3x3) x 5

That's the only way to make sure that 16 and 9 go into it.

Bob

hi paulb203

They mean the same. In number theory, division is defined as the inverse operation to muliplication.

So if a times b = c then c divided by b is a.

This is why division by zero has no meaning.  [a times 0 = 0 for all a, so if we tried to calculate 0 / 0 we'd get a where a can be anything] and  [ x / 0 would mean what number must we multiply 0 by, to get x as our answer.  This cannot be answered.]

So an example would be what number when doubled gives 10 ?  x times 2 gives 10 can then be re-written as x = 10 / 2

Bob

hi woodturner550

I'd be interested to see some output from your program.

In Python,  time.time() returns a UTC value ; in seconds and decimal fractions of a second. A computer's motherboard must save and increment this value using the computer's internal clock.  As the motherboard keeps running even when the computer is switched off this will get out of step with the standard time and so the operating system must, from time to time, log in to a standard time somewhere to get the current UTC.  This will take time (no pun intended) so the computer's version of UTC will never be exact. But that doesn't matter for your purposes, in fact, it helps because your version of UTC will be different from anyone else's.

Do you know how, for example, MS Excel makes up it's random numbers. Could it be that it already uses something similar to what you are proposing ?  If it does then MS probably already has the copyright on this.  It may also be that they don't publish this coding;  precisely because they don't want it hacked.

If a computer virus were able to modify the system's record of UTC, could this be a weakness that would allow a hacker to predict your numbers ?  If you can devise a way to be certain that the internal value ( of UTC) held by the computer's operating system is genuine; then you've got a way to be certain your method is reliable.

I think there are statistical tests for the reliability of randomness.  You'd need a large set of random outputs.

We do have a computer section called Coder's corner.  I can move this thread across if you would like.

Bob

hi woodturner550

Welcome to the forum.

I've used the 'random generator' on Excel quite a few times and the BBC Basic one, and I've never thought to question how it's done. Each time you 'call' the function it yields a different value so it looks random. But, you're right, it must be using a formula and hence results could be predicted by someone who knows how it works.  Many years ago in the UK the Premium Bond organisation made a piece of equipment which they called ERNIE (= Electronic Random Number Indicator Equipment).  Folk who buy premium bonds don't get a percentage interest, rather their numbers are entered in a draw, which pays out money to the lucky winners.

As far as I'm aware it has never been 'hacked'.

All digital computers run from an internal clock.  If you can find out how to access this then you would have your stop watch moment. Since the clock cycles are very fast it would be very hard for anyone to predict the moment you did this.

Subsequent numbers could be generated from within your routine by the same means.  The length of time that would elapse between numbers would probably be indeterminate too as the computer is doing thousands of other things whilst running your routine, which would be taking an unpredictable amount of time each, so the clock cycles would have progresed by an indeterminate amount each time.

Bob