Question

Help with related rates

Answer 1

Machine Design: The endpoints of a movable rod of length 1 meter have coordinates (x,0) and (o,y). The position of the end on the x-axis is:

x(t)=3/5 sin(pi t)

a. Find the time of one complete cycle of the rod.

b. What is the lowest point reached by the end of the rod on the y axis.

c. Find the speed of the y-axis endpoint when the x-axis endpoint is (3/10,0)

Answer 2

@Loser66

Answer 3

They don't mention the units that time is being measured in

but to find the period I would match up
\[ \sin \left( \frac{2 \pi}{T} t\right) \]
with your sinusoid
in other words, we match
\[ \frac{2\pi}{T}= \pi \\ T= 2\]
in whatever units they are measuring time.

Answer 4

b. What is the lowest point reached by the end of the rod on the y axis.

that happens when the "bottom end" on the x-axis is as far along the x-axis as it can go.

Answer 5

what is the largest number you can get from
x(t)=3/5 sin(pi t)
?

You start by knowing what is the largest value sin can ever be ?

any idea?

Answer 6

Sorry was gone for a bit

Answer 7

the largest value sin can be is 1 @phi

Answer 8

so what is the largest that x(t) can be ?
x(t)=3/5 sin(pi t)

Answer 9

you put in 1 for sin(pi t) (i.e. make this as big as it can be)
and you get
x= 3/5 * 1
or just
x= 3/5

Answer 10

So it doesnt matter that t is in sin(pi t)?

Answer 11

I hope it's clear that the more the rod slides to the right, the lower it must slide down

they do not ask when y is smallest, just what is the value of y. so we don't care about what t it is when sin( pi t) =1, we just care that the biggest sin can get is 1, and that means the biggest x can be is 3/5

Answer 12

but you still have to find y
I would use pythagoras

Answer 13

Ohhh I see yes that makes sense

Answer 14

so for pythagoras we can use 3/5^2 + b ^2 = 1^2?

Answer 15

yes

Answer 16

y= sqrt(1-.12)

y= . 93

Answer 17

try again

Answer 18

y^2 = 1^2- 3/5^2

Answer 19

it's (3/5)^2 not 3 divided by 5^2

Answer 20

oh brackets

Answer 21

y= .6?

Answer 22

y^2 = 25/25 - 9/25 = 16/25

Answer 23

oh its the equivalent of 4/5

Answer 24

part c looks like calculus. are you studying calculus ?

Answer 25

Yes I am

Answer 26

you should start by writing down the pythagorean equation for this problem
y^2 = 1 - x^2

now take the derivative of both sides with respect to x
can you do that ?

Answer 27

I meant with respect to "t" (not x)

Answer 28

so y^2(dy/dt)=1-x^2(dx/dt)

Answer 29

almost. d/dt y^2 = 2y dy/dt
and ditto for the x^2

Answer 30

2y= 2x since 1(d/dx) = 0

Answer 31

ok, getting closer

2 y dy/dt = -2 x dx/dt

Answer 32

we can cancel the 2's

y dy/dt = -x dx/dt

we need to find some numbers. but we know x is 3/10
we need y and we need dx/dt

to find dx/dt we start with

x = 3/5 sin (pi t)
and take the derivative with respect to t

Answer 33

Question what exactly does dy/dt and dx/dt representing ? Usually its some rate, but I am confused on this one

Answer 34

y and x are "linked" (if one changes the other changes)
and x changes with t (remember x= 3/5 sin(pi t)
so x changes with t at a rate dx/dt
and y also changes with t at a rate dy/dt

Answer 35

In the real world, as bottom of the rod slides along the floor , it goes at some speed, also known as dx/dt

Answer 36

so we have y (dy/dt) = -(3/5 sin(pi t)) (dx/dt)

Answer 37

I would not do that. we know x is 3/10 so we should not use the 3/5 sin (pi t) instead ... just use 3/10

on the other hand we do have to find dx/dt
so we have to start with
x= 3/5 sin(pi t)
now find the derivative with respect to t

Answer 38

So wait
like this?:

3/10 = 3/5 sin(pi t) (dx/dt)

Answer 39

this is a bit complicated. so you have to go slow.

do this
\[ \frac{d}{dt} x = \frac{3}{5}\frac{d}{dt} \sin(\pi t) \]

Answer 40

oh you were saying use 3/10 for the other equation

Answer 41

\[\frac{ d }{ dt }(\frac{ x }{ \sin(\pi t) }) = \frac{ 3 }{ 5 }\]

Answer 42

Does this look like the proper set up?

Answer 43

no, that complicates it.

d/dt of x is dx/dt

Answer 44

Hmmm I am a bit confused, but I assume its just dx/dt because its just x

Answer 45

yes. x is a function of t, and the change in x with respect to t is dx/dt
so you have
\[ \frac{dx}{dt} = \dot{x} = \frac{3}{5} \frac{d}{dt}\sin \pi t\]

Answer 46

sometimes people show the derivative with respect to time as the variable with a "dot" over it.

Answer 47

Oh so now we can just plug that into our equation we got earlier?

y dy/dt = 3/10(3/5(d/dt) sin(pi t)) ?

Answer 48

I would first find the derivative of sin pi t

Answer 49

I have not dealt with derivatives of trig functions yet. Calc 1 at my school is just dealing with just algebra it seems :/

Answer 50

I assume based on my past knowledge it will have something to do with cosine

Answer 51

d sin x = cos x dx

or in this case
\[ \frac{d}{dt} \sin \pi t = \cos \pi t \frac{d}{dt} \pi t \\
= \cos (\pi t) \pi \frac{d}{dt} t \\
= \cos (\pi t) \pi \\
= \pi \cos (\pi t) \]

Answer 52

d/dt of t is dt/dt = 1

Answer 53

Yes that makes sense looking at it

Answer 54

so all of the means
\[ \frac{dx}{dt} = \pi \cos \pi t\]

but we need a number for the cosine part

we do know
\[ x = \frac{3}{5} \sin \pi t\\
\frac{3}{10}= \frac{3}{5} \sin \pi t\]

Answer 55

that means
\[ \frac{dx}{dt} =\frac{3}{5} \pi \cos \pi t \]

Answer 56

So now this is ready to get plugged back into our equation?

Answer 57

using the info
\[ x = \frac{3}{5} \sin \pi t\\
\frac{3}{10}= \frac{3}{5} \sin \pi t \]
we want to find what pi t is
i.e. what angle is that ?

Answer 58

if you simplify that you get
sin pi t = ½

Answer 59

Is it pi/3?

Answer 60

since pi/3 = 1/2, sqrt3/2

Answer 61

sin 30º = ½

that is pi/6
but the important thing is cos 30 = sqr(2)/2
so we can replace
cos pi t with sqr(3)/2

Answer 62

and fyi, t = 1/6 : sin pi t = ½ ---> pi t = pi/6 ---> t= 1/6
we don't need to know that, but it's good to be able to find t.

Answer 63

I am confused how we can replace cos pi t with sqrt 3 /2

Answer 64

does this make sense
\[ x = \frac{3}{5} \sin \pi t\\
\frac{3}{10}= \frac{3}{5} \sin \pi t \\
\frac{1}{2} = \sin \pi t \\
\sin \pi t = \frac{1}{2} \]

Answer 65

we used
x(t)=3/5 sin(pi t)

and put in 3/10 for x

Answer 66

you then take the inverse sin (or "arcsine") of both sides
\[ \sin^{-1}\left(\sin \pi t \right) = \sin^{-1} 0.5 \\
\pi t = \frac{\pi}{6}
\]

that means when t= 1/6, x is at 3/10
(of course this rod is moving back and forth... we are picking the first time starting from time 0)
at that time t= 1/6 cos(pi t) is cos(pi/6) which is sqr(3)/2

Answer 67

now we have
\[ \frac{dx}{dt} =\frac{3}{5} \pi \cos \pi t \\
\frac{dx}{dt} =\frac{3}{5} \pi \frac{\sqrt{3}}{2} \]
or
\[ \frac{dx}{dt} = \frac{\pi\ 3 \sqrt{3}}{10} \]

Answer 68

This is starting to get really confusing to me the more we get into it

Answer 69

yes, it's complicated. I am surprised how messy it is, based on parts a and b being relatively simple.

Answer 70

Yes That is why I am confused especially since I have not dealt with differentiating trig things before

Answer 71

Even you can't do all the parts, what you did was write
\[ y^2 = 1 - x^2\]
(pythagorean theorem with a bar of length 1)
you then found the derivative with respect to "t" (a third variable)
we are using "implicit differentiation" which you may not have learned?

you get
\[ 2y \frac{dy}{dt} = - 2 x \frac{dx}{dt}\\
\frac{dy}{dt} = - \frac{2x}{y} \frac{dx}{dt} \]

Answer 72

correction: \[ \frac{dy}{dt} = - \frac{x}{y} \frac{dx}{dt} \]
we want dy/dt
we have the formula, but we need numbers for x , y and dx/dt

x is easy, they tell us x= 3/10

Answer 73

I do not believe I have learned

Answer 74

Could we put in 3/10 to find y? in the Pythagorean thereom

Answer 75

now we need y and dx/dt

y you can find using
y^2 = 1 - x^2
and putting in x= 3/10
you get the ugly value
\[ y = \frac{\sqrt{91}}{10}\]

Answer 76

So now we need to find dx/dt but didnt we find that it equals pi 3sqrt3/10?

Answer 77

yes

Answer 78

so dy/dt = -(3/10/(sqrt91/10))(pi3sqrt3)/10

Answer 79

yes, but we can clean that up a little. It's an ugly number, unless they want it as a decimal rounded to a few digits.

Answer 80

I'm getting about - 0.513

Answer 81

I got -1.7 hmmm

Answer 82

it might be easier to change all the factors to decimals
for example y is about 0.9539
x is 0.3
dx/dt is 1.6324

Answer 83

ahh I see so the speed of the y axis endpoint is 1.6324