OT: Mathematical Conundrum III

[Cursitor Doom]

In the 3rd of my series on ever-so-slightly-getting-more-complicated mathematical conundra, I pose a problem based on two road users:

A motorist and a cyclist start off together from a point 'A' to travel on the same road to point 'B' which is 18 miles away. The motorist starts at a uniform speed 15mph faster than the cyclist. When the motorist has got halfway, he is delayed for 30 minutes and thereafter travels at a speed

20% less than his original speed. He reaches 'B' 15 minutes before the cyclist. Find the cyclist's speed.

[David Lang]

Does the cyclist ride on pavements, ignore one way systems, jump traffic lights etc?

:-)

[The Natural Philosopher]

Is the cyclist speed constant? Does he get a puncture, take a short cut through a pedestrian precinct, or put his bike on the train?

[Huge]

Oh, do bugger off. :o)

'O' Level maths was 45 years ago. Who can be arsed with that sort of thing?

Tell me how to calculate how many more boxes I have to buy to fit the contents of my house into.

[Tim Streater]

12mph.

Dear oh dear I must be getting old. I had the method right but with three major howlers during the calcs I'd fail O-level these days.

[Chris French]

Does the motorist jump red lights, break speed limits, ignore people waiting at zebra crossings, cut up others etc. ?

:-)

[Cursitor Doom]

Well done, Tim. However, since you haven't shown your working, we don't know if you got the correct solution via nefarious means... ;->

[NY]

Zero marks - you didn't show your working :-)

[Cursitor Doom]

We're not venturing into calculus territory here *yet*, so you have to assume the speeds achieved are attained *instantly* with no acceleration from rest taken into account. This would, of course, kill both participants in real life. ;-)

[Tim Streater]

I started with:

m = car's initial speed (mph) (27mph) c = cyclist speed (mph) (12mph) t = car elapsed time (hours) (1.25hrs)

Car:

9/m + 0.5 + 9/0.8m = t

Bike:

18/c = t + 0.25

Speeds:

m = c + 15

giving a quadratic equation in m. Easy enough in principle but I still made 3 mistakes as I went along.

In questions like this, usually the answer is a whole number, so I suspected my first speed for the car (31.2xxxxxx mph) was probably wrong. Second and third ones were whole numbers but made the bike travel at 42 or 39 mph (implausible). In any case plugging the numbers back in to the original equations didn't work until I got the right answer.

[The Natural Philosopher]

And remember to allow for the clock running slower for the car, because it travels faster.

Not to mention the effects of gravity if the road is not flat.

[Richard]

Yes, but less frequently than the cyclist because the motorist gets fined for the stuff that the cyclist doesn't.

[Tim Watts]

That's easy - 3 times as many as you first thought. However, packing is a non linear function by volume of goods: you will spend the first 25% of the time packing 75% of the boxes with the big things, books, plates, etc.

The remaining 25% of "stuff" will take more or less forever, asymptotically approaching zero remaining.

[Huge]

I fear you are right.

[Tim Watts]

It seemed to be the case when I last moved. The movers are good at guessing numbers of boxes so I was supplied with "a lot".

We thought it was going super well when we'd used half of them in a couple of days. But of course, we were filling each one with books, a few big things and crockery which is a plate wrapped with a sheet or two of tissue paper.

After all the big stuff was done, you get bits of stuff in drawers, shelves, cupboards - it all tends to need tissue paper and it occupies sod all of a box, so now 1 box takes an hour to three to fill.

My advice - get ruthless with purging before starting!

[John Rumm]

Last time we moved, it was a double move - contents of mother's large Victorian 4 bed semi (approx 1800 sq ft), the movers said 75 boxes required.... Then they looked at our by then (i.e. with loft conversion)

1300 sq ft 5 bed semi, and said 200 boxes. I thought they were being silly. They even agreed a refund if we needed fewer boxes. I thought about that as we packed to 200th and still had stuff for another 4!

[Chris French]

Best thing we did on the last move was get the movers to do the final packing. as an extra on top of a 200 mile move it wasn't a lot really

We sorted stuff out, I did all the outside stuff, garage, greenhouse, loft etc. and packed some of the inside stuff we wouldn't need..

On the day they came in, packed all the kitchen, breakables, everything else left. Much less stressful. They also packed some stuff in a different way than i would have. Egg plates went into one of those big square moving boxes. But they use lots and lots of the paper scrunched up to separate stuff, rather than wrapping and stacking in carefully into a smaller box.

nothing got broken.

[Dave Baker]

Let the car's initial speed be X mph and the bike's be X-15 mph.

Car journey time in minutes is 60(9/X) + 30 + 60(9/0.8X) Bike journey time is 60(18/(X-15))

Car journey time = 540/X + 30 + 675/X Bike journey time is 1080/(X-15)

540/X + 45 + 675/X = 1080(X-15)

multiply both sides by (X-15)

540-8100/X + 45X - 675 + 675 - 10125/X = 1080

45X - 18225/X = 540

multiply by X

45X^2 - 18225 = 540X

divide by 45 and subtract sides

X^2 - 12X - 405 = 0

(X + 15).(X-27) = 0

Car initial speed is either -15 mph or +27 mph. Clearly the latter. Bike speed is 27-15 = 12 mph

I'm not sure why we're doing such trivial quadratics in here but it occupies a few minutes I suppose.

[Clive George]

I've heard enough people say similar things that even I'd consider it if I was moving house, and I'm a tight bastard.

OTOH DIY is on topic for this group, and probably means you get a better purging job.

[Nige Danton]

I move home (and country) fairly regularly and no matter how ruthless I am before each move we still ship stuff that isn't needed any more e.g I've got unopened boxes from 2 moves ago.