Question

PLEASE HELP WITH THIS CALCULUS PROBLEM!



A carousel has a diameter of 16m and completes a revolution every 30s.

a) Model the north-south position of a rider on the outside rim of the carousel using a sine function.

b) Differentiate the function in part a)

c) Determine the north-south speed of the rider.

d) What is the position of the rider on the carousel when this maximum north-south speed is reached?

PLEASE SHOW WORK!

Answer 1

a) This is uniform circular motion but we can model the x motion and the y motion separately. Calling y>0 north and y<0 south,
\[y=8\sin wt\]where t=time in seconds and w=angular velocity in radians/second. The 8 appears in the question because the north-south position varies between +8m and -8m since the diameter of the wheel is 16m. The angular velocity is given by:\[w=\frac{2pi}{30}=\frac{pi}{15}\]so we have:\[y=8\sin (\frac{pi}{15}t)\]

Answer 2

Ok, thank you for that, now to part b

Answer 3

give part b a try and let me know if you get stuck

Answer 4

I dont know how to differentiate it

Answer 5

do you know how to differentiate \[y=\sin x\]?

Answer 6

cosx

Answer 7

agreed

Answer 8

now y=8sinx just differentiates to 8cosx so we're almost there.

Answer 9

so its just 8cosx(pi/15)(t)?

Answer 10

not quite; we need to apply the chain rule to this one

Answer 11

\[\frac{d}{dt}8\sin (\frac{pi}{15}t)=8\cos(\frac{pi}{15}t)*\frac{d}{dt}(\frac{pi}{15}t)\]

Answer 12

so is it: 8/15picos(pi(t)/15)

Answer 13

yep

Answer 14

are you sure?

Answer 15

\[dy/dt=\frac{8pi}{15}\cos (\frac{pi}{15}t)\]yeah

Answer 16

so how about part c)?

Answer 17

Well part c is badly worded. Do they want the value of the maximum speed? The north-south speed is given by \[\left| dy/dt \right|\] which varies with time. So the north-south speed is just the absolute value of the function in part b.

Answer 18

im not sure, thats how its written

Answer 19

i think its asking for maximum according to part d)

Answer 20

my answer would be \[\left| \frac{8pi}{15}\cos (\frac{pi}{15}t) \right|\]Now the largest the absolute value of the cosine of anything can be is 1...this occurs when t=0 and t=-pi and t=2pi. This max speed is equal to (8*pi)(15)

Answer 21

are you sure thats the answer?

Answer 22

If you see an error, let me know

Answer 23

ok, onto part d)

Answer 24

which equation gives us the position of the rider?

Answer 25

Im not sure

Answer 26

btw, I got 377 for (8*pi)(15). Whats the unit?

Answer 27

ok, we already came up with it...it is precisely the equation asked for in part a. Now, we know the values of t for which the north-south velocity is at a maximum so all you need to do is plug in these value of t into the position function to answer this.

Answer 28

units are in m/s

Answer 29

I dont understand what values of t you are talking about

Answer 30

and (8*pi)/15 is about 1.68...not sure where you got 377?

Answer 31

"This max speed is equal to (8*pi)(15)"

Answer 32

oh, that's a typo on my part...should read (8*pi)/15

Answer 33

ok so what values of t?

Answer 34

"Now the largest the absolute value of the cosine of anything can be is 1...this occurs when t=0 and t=-pi and t=2pi. This max speed is equal to (8*pi)(15)
"

Answer 35

so I just substitute all those values and thats the answer?

Answer 36

substitute them into the equation from part a, and yes, those should be the answers

Answer 37

when writing pi, is it 3.14 or 180?

Answer 38

3.14 radians

Answer 39

so when t = 0, position = 0?

Answer 40

the y position equals zero, yep

Answer 41

and when t= -pi, y= -0.9186?

Answer 42

no...when t=-pi, y=0 again

Answer 43

how?