Maths Ext 1 help please (1 Viewer)

[plshelp]

Ok question is from Cambridge 3u yr 12 textbook: Chapter 7A Applications of APs and GPs

I dont know how to start...

Full working solutions would be nice...or just explain it to me...

Sorry for my dumbness...

17) Two bulldozers are sitting in a construction site facingeach other. Bulldozer A is at x=0, and bulldozer B is 36 metres away at x=36. Abee is sitting on the scoop at the very front of bulldozer A. at 7:00am theworkers start up both bulldozers and start them moving towards each other atthe same speed V m/s. The bee is disturbed by the commotion and flies at twicethe speed of the bulldozers to land on the scoop of bulldozer B.
a) Show that the bee reaches bulldozer B when it is at x=24

b) Immediately the bee lands, it takes off again and fliesback to bulldozer A. Where is bulldozer A when the two meet?

c) Assume that the bulldozers keep moving towards each otherand the bee keeps flying between the two, so that the bee will eventually besquashed.
i) When will this happen?
ii) How far will the bee have flown?

 

[kooliskool]

Hi, this question I agree it's hard to know where to start, but the best way to start is to draw up the diagram, and it shouldn't be too hard after that.

So for a), I assume you know it already? Since the bee started together with bulldozer A on one end with velocity 2V m/s, it will be completing the journey with bulldozer B coming from the other end at velocity V m/s. So they will both complete 36 metres at 3V m/s, Which will take them 36/(3V)=12/V seconds. Hence the bee travelled:

metres.

b) Now here is the tricky bit, you should consider that the journey is shortened, but the exact scenario is reversed. Since the bulldozers A and B are traveling at V towards each other, the journey is shortened from both side by:

metres

Hence A is going to the right from x=12 and B and the bee are going to the left from x=24. Again, since the bee is completing the journey with B together this time, (the total speed of completing it is 2V+V=3V), they will meet after (36-24)/3V=4/V seconds, so they will meet at:

metres.

c) Here is the serious about this part and is the part which makes this a series (or a sequence).

i) If you gathered from previous parts, each time the journey is shortened by 2/3 of its previous journey, so only 1/3 left, and the bee and the bulldozers complete them at the same speed. Hence the time taken each time is 1/3 of the previous time taken, which can be seen from part a) and b).

Hence, the total time until the bulldozers squash the bee is an infinite series of:



seconds

This makes sense since it will only end when the bulldozer finish the journey and meet each other half way in the middle right?

ii) Also gathered from previous parts, if you looked carefully, the bee always complete 2/3 of the shortened journey (realise this is a constant ratio of the journey left, so makes no difference to the overall formula below). But each time the journey is shortened by 1/3, hence the sum of the distance the bee flown is an infinite series of:



metres

This is pretty weird for a result I know, that means the bee has flown across the whole journey, but it makes sense if you think about it. The bee is traveling at 2V m/s the whole time, so isn't it the bee has traveled twice the distance the bulldozers have traveled? Which means the bee traveled 36 metres while the bulldozers only traveled 18 metres.

This can also be proved by: the whole journey is taking 18/V seconds to complete, the bee is travlling at 2V m/s, so the bee has travelled 18/V*2V=36 metres altogether.

Sorry if I didn't make sense somewhere in the solution, do post any questions again if I'm not clear at some part and good luck