Bubble's specialist questions

[bubbles]

What I have done so far:


How do you do 1. c) (i) ?

[cara.mel]

In b) the volume is the integral from 0 to 30, not from -10 to 10, as you are rotating around y axis

For c i), I would guess do the integral again but from 0 to h, then differentiate it. There is probably an easier way and/or what I have said is wrong

[Mao]

question 3:
















given (0,30) [very messy...]



hence, when h=0, t...

?!

[i still have a feeling its wrong...]

[Mao]

yes, of course. thank you

it still feels too whacky
haha

[shinny]

It's a pretty ugly answer but it sorta sounds about right given that the rate of change of volume is in proportion to its height. So as it gets lower and lower, the rate of change gets infinitely small and the last few drops will take quite a while to clear. I'll verify the answer on CAS afterwards.

Now for the proper working out for Q2.






















EDIT: yep, checked with CAS and theres no algebra mistakes in your working out. Pretty sure its the right answer then as there's no fault in your method I'm pretty sure =\

[bubbles]

yep i get it now, thank you guys!
Nearest minute therefore (47606.62s) / 3600 = 13 hrs and 13 minutes
I'll confirm the answer to c) iii. once i get a hold of it sometime this week

[bubbles]

?!

[i still have a feeling its wrong...]

yep that's the correct answer in seconds

[bubbles]

1d) I don't have the question with me but from memory: Given =
= found in part a) form a new rate of growth equation.
Any ideas?

[Mao]

initially looking at it, I don't see anything especially wrong with your workings [or i might have overlooked something]. So heres my workings:







    partial fractions

   

   

   





given



as required

1d) plug numbers into

[Mao]

okay, found your problem, it was when you tried to work out the partial fraction
what should have happened is:



which is the negative of your expression

[bubbles]

~kinematics
(I dropped physics this year and now it has come back to haunt me -___- )

An object moves in a line with velocity v m/s given by . The object starts from the origin. Find:
b) the maximum velocity
max velocity occurs when v'(t) = 0
differentiate v via quotient rule, therefore t=1
sub t=1 into v(t), hence = 0.5m/s
c) the distance travelled in the third second
?
d) the position-time relationship
how do you anti-differentiate again ?

[enwiabe]

v = dx/dt = t/(1+t^2)

You want the time travelled in the third second, so integrate t/(1+t^2) from t = 2 to t = 3 (I.E. the third second).

I'll give you a hint for the integral of t/(1+t^2) - it looks eerily like f'(x)/f(x), eh?

Also, you don't need Physics to do well in the Physics part of spec! Specialist Maths goes WAY above and beyond Physics in terms of kinematics, so don't sweat it.

[cara.mel]

You want the time travelled in the third second

I predict this is 1!

[bubbles]

I'll give you a hint for the integral of t/(1+t^2) - it looks eerily like f'(x)/f(x), eh?

Thanks enwiabe i understand a) but I haven't yet grasped how to integrate t/(1+t^2).
okay this is going to expose how inept i am with specialist maths  :



therefore ??

The answer is: where does the half come from?

Using the correct answer at the back:
c)
 
    (which is the correct answer)

[cara.mel]

Make the substituion

=>



Subbing this in to the equation



Then you can cancel the ts and finish it off and you see how the 1/2 ends up being there.
I would advise always using substitutions and writing out all the working vs trying to be clever and do it in your head, I stuff up too many times doing that