# Math

posted by .

Hi, this is a question I posted a couple of days ago and answers below.I am still really confused and unfortunately can't copy a picture into this. I thought it would be y-x but have no idea how to calculate that, but as the pistons move in towards each other, wouldn't the distance have to be 1/2 (y-x)? This is Year 11 maths in Australia. Any further explanations would be really appreciated. Thankyou to Dr Reiny and everyone else who has responded.
Melody

Two pistons A & B move backwards and forwards in a cyclinder. Distance x cm of the right hand end of piston A from point 0 at time t seconds is x=3sin(2t)+3 and the distance y cm of the left hand end of piston B by formula y=2sin(3t-pi/4)+8.5
The pistons are set in motion at t=0
How do I work out the minimun distance between the pistons and the time it occurs? I am really lost on this, can you please help.
Thanks

Maths - Reiny, Wednesday, October 14, 2009 at 8:53am
Looks like a rather nasty question
I see the distance (d) as
d = 3sin(2t)+3 + 2sin(3t-pi/4)+8.5
= 3sin(2t)+ 2sin(3t-pi/4)+ 11.5

I first ran this through a primitive graphing program I have, and it showed the above to have a period of 2pi, and a minimum value roughly between t = 1.5 and t = 1.7

to get exact answers is a messy Calculus problem
dd/dt = 6cos(2t) - 6cos(3t-pi/4)
= 0 for a max/min of d

So we have to solve
6cos(2t) - 6cos(3t-pi/4) = 0 or
cos(2t) - cos(3t-pi/4) = 0
the trouble is that we need to have our trig functions contain the same period.
we know cos 2t = 2cos^2(t) - 1
and cos(3t - pi/4) = cos3tcospi/4 + sin3tsinpi/4
= (1/ã2)(cos3t + sin3t)

now there are expansion formulas for
cos 3x and sin 3x in terms of cos x, but I have a feeling that the question couldn't possibly be that messy.

Are you using graphig calculators?
What level is this?

Math experts needed here. - Reiny, Wednesday, October 14, 2009 at 9:00am
Can somebody see through the above mess?
Am I on the wrong track?

Maths - bobpursley, Wednesday, October 14, 2009 at 9:36am
I see the distance between the pistons as y-x. It is so hard to see without a diagram.

d= 2sin(3t-pi/4)+8.5 -3sin(2t)+3
Then proceed as in Dr Reiny's solution.

In past days, we would have put this on the analogue computer and ran a graph for few seconds.

Maths - MathMate, Wednesday, October 14, 2009 at 12:11pm

I started from Reiny's result
cos(2t) - cos(3t-pi/4) = 0
which I validated as correct.

By equating the two cosines, I conclude that:
2t + 2kƒÎ = 3t-ƒÎ/4
after adding 2kƒÎ for generality, which gives
t = ƒÎ/4 + 2kƒÎ
Numerical solutions are: ... -5.5, 0.78, 7.07, ...
Since cosine is an even function, one of the arguments of cosine could be negative, therefore
-2t + 2kƒÎ = 3t - %pi;/4
which gives
t=(1/5)(ƒÎ/4+2kƒÎ)
Numerical solutions are: ... -6.12, -4.86, -3.61, -1.1, 0.157, 1.41, 2.67, 3.92, 5.18, 6.44 ...

To MathMate - Reiny, Wednesday, October 14, 2009 at 1:51pm
Of course!
How simple was that step, eh?

And here I was about to enter the mathematical hinterland.

Thank you!

To Reiny - MathMate, Wednesday, October 14, 2009 at 2:43pm
That's what happens when you know a lot!

Maths - bobpursley, Wednesday, October 14, 2009 at 4:14pm
Nice work.

• Math -

Melody, the difference is indeed y-x, which is a short form for f(t) = y(t)-x(t).
Since they both vary with time in a periodic manner, it is important to find out when the f(t) attains the maximum, and what the value of the maximum is.
This graph shows exactly the function
f(t) = y(t) - x(t) as a function of time:
http://img62.imageshack.us/i/1255515491func.png/
You will notice peaks and valleys at different times t.
To find the maximum (and minimum), Dr. Reiny has evaluated the derivative of f(t) with respect to t and equated f'(t) to zero.
The solutions where t is a maximum or minimum have been given as:
t1 = ƒÎ/4 + 2kƒÎ
t2 = (1/5)(ƒÎ/4+2kƒÎ)
and k is an integer constant.
By substituting different integral values of k into the above solutions, we obtain values of t for which the function f'(t)=0, or where f(t) is a maximum or a minimum.
These values have been evaluated as:
t1 = -5.5, 0.78, 7.07, ... and
t2 = -6.12, -4.86, -3.61, -1.1, 0.157, 1.41, 2.67, 3.92, 5.18, 6.44...
respectively.
Now if you would go back to the graph of f(t) and you can pick out the minimum values of about -9.545 at t=2.67, corresponding to (1/5)*(π/4+4π).
You can evaluate y(t)-x(t) at different locations where f'(t) = 0 to make sure you get the global maximum. The minimum distance can be read from the graph and the exact time when it occurs can also be calculated from the solutions t1 and t2 above.
I hope that explains what you need to know, but do not hesitate if you need more details.

Final note: it would be much clearer if you study the graph of the function y(t)-x(t) which is at:
http://img62.imageshack.us/i/1255515491func.png/

The graph of the derivative is shown in :
http://img379.imageshack.us/i/1255515491deriv.png/

• Math -

Note:
the minimum distance occurs at
t=(1/5)(1/4+6)π
where
f'(t)=0 and
f(t)=0.5

• Math -

Also, please check the signs, I believe y(t)-x(t) is always positive. The graph shown is x(t)-y(t). My apologies.

## Similar Questions

1. ### Math

Can someone walk me through these becuase I'm just not getting it.... Let f(x)= 3 x 4^x. A.) 16f(x)= f(x+2) B.) f(x)= f(x-1) The quickest way to answer this question is to try different test values for x. My favorite was the number …
2. ### English

I asked this question a couple of days days ago and noone was really sure. "Let me in," Said the pin. Which is this an example of: 1. assonance 2. a metaphor 3. alliteration 4. personification I know it's either assonance or alliteration.
3. ### Grammar

I'm still really confused with this one; I posted it up a couple of days ago but never got the answer Is this a dangling modifier?
4. ### poems

for english class i have to write a poem personifying something. I chose to personify a picture in a picture frame. I am not at all good at poems and was wondering if you guys could read mine and tell me what changed i should make …
5. ### Math

I posted these 2 questions a couple of days ago and although the help I received would have been more appreciated if it didn't come with an underhanded comment, I did see where I made my mistake with the first question, however I would …

No one was answering my questions for the first question I posted so I posted it again. answers for 12345: 1. 9 3/5 2. 4 1/2 3. 4 3/4 4. still 7 6/9 5. still 10 16/18
7. ### Site not working correctly/Physics

I posted a question a couple of days ago which was answered by DRWLS - thank you. Unfortunately I am unable to read the guidance provided as the typing is obscured by the adverts down the right hand side of the page. Look forward to …
8. ### Math 1 tiny question

I posted this question a couple days ago and I also got help but I have one more question , so is the final answer to the question below: 30 downloads would make site A a better deal. No more than 30 and no less or is it equal to 30 …
9. ### Pre-Calculus

I posted this question about an hour ago, got a response but still confused. Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to …
10. ### Math

Hello, I'm not sure how to do this question. Question A wheel with radius 5 cm is being pushed up a ramp at a rate of 4 cm per second. The ramp is 620 cm long, and 180 cm tall at the end. A point P is marked on the circle as shown …

More Similar Questions