Expert: Paul Klarreich - 10/23/2006

Question
how would i differentiate (1/sinY)(ysin(theta))?!?! i know it involves the chain rule, but i'm unsure how to carry that out in this problem


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The text above is a follow-up to ...

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Thank you so much for your help!!! I feel like I'm beginning to really grasp the concept.  I do have a couple questions about number 3:
3)As sand leaks out of a hole in a container, it forms a conical pile whose height is always equal to the radius of the pile.  If the height of the pile is increasing at a rate of 6 inches per minute, find the rate at which the sand is leaking out of the container when the height of the pile is 10 inches.

I think the formala that will relate all the variable together is V = pi/3 * r^2 * h, but i'm not positive if the rate of change of the volume of the pile is equal to the rate at which the sand is leaking out of the container.

And on number 4, I am completely confused :(


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The text above is a follow-up to ...

-----Question-----
Studying related rates

1)As a circular steel griddle is being heated, it's diameter changes at a rate of .01 cm/min.  Find the rate at which the area of one side is changing when the diameter is 30 cm.
2)A girl starts at a point A and runs east at a rate of 10 ft/s.  One minutes later, another girl starts at a point A and runs north at a rate of 8 ft/s.  At what rate is the distance between them changing one minute after the second girl starts?
3)As sand leaks out of a hole in a container, it forms a conical pile whose height is always equal to the radius of the pile.  If the height of the pile is increasing at a rate of 6 inches per minute, find the rate at which the sand is leaking out of the container when the height of the pile is 10 inches.
4)A model rocket enthusiast launches a rocket vertically from a point that is 5 miles away from his tracking device.  Assume the launch site and the tracking device are at the same elevation.  For the first 20 seconds of flight, the angle of elevation of theta changes at a constant rate of 2 degrees/s.  Find the velocity of the rocket in miles per hour when the angle of elevation is 30 degrees.


-----Answer-----
Questioner:  Rachel
Category:  Calculus
 
Subject:  Related Rates
Question:  Studying related rates

1)As a circular steel griddle is being heated, it's diameter changes at a rate of .01 cm/min.  Find the rate at which the area of one side is changing when the diameter is 30 cm.
2)A girl starts at a point A and runs east at a rate of 10 ft/s.  One minutes later, another girl starts at a point A and runs north at a rate of 8 ft/s.  At what rate is the distance between them changing one minute after the second girl starts?
3)As sand leaks out of a hole in a container, it forms a conical pile whose height is always equal to the radius of the pile.  If the height of the pile is increasing at a rate of 6 inches per minute, find the rate at which the sand is leaking out of the container when the height of the pile is 10 inches.
4)A model rocket enthusiast launches a rocket vertically from a point that is 5 miles away from his tracking device.  Assume the launch site and the tracking device are at the same elevation.  

For the first 20 seconds of flight, the angle of elevation of theta changes at a constant rate of 2 degrees/s.  Find the velocity of the rocket in miles per hour when the angle of elevation is 30 degrees.

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Hi, Rachel,

That's a LOT of problems.  It sounds as if you are just starting R-R problems, so here is my standard introduction:

1. Identify the variables in the problem -- the things that change.  Give them names.

2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.

3. Determine a relationship (yes, it is called 'related rates' for a reason) between the

variables.  Use a diagram, use your life experience, use your general knowledge and brilliance,

whatever you have to.  This is the key step.

4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the

unknown rate.


Your examples:

1)As a circular steel griddle is being heated, it's diameter changes at a rate of .01 cm/min.  

Find the rate at which the area of one side is changing when the diameter is 30 cm.

Variables:
D = diameter of the griddle.
a = area of one side.

Rates:
dD/dt = rate of increase of diameter, GIVEN as 0.01 cm/min
da/dt = rate of increase of area, TO BE FOUND.

Relationship:

We know the formula for area of a circle:

a = pi r^2,  and  r = D/2,  so this would be  a = pi D^2/4

Differentiate:

da/dt = 2pi D/4  dD/dt

da/dt = pi D/2  dD/dt

Substitute:  D = 30,  dD/dt = 0.01

da/dt = pi(30)/2 (0.01) = pi 15(0.01) = 0.15 pi

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2)A girl starts at a point A and runs east at a rate of 10 ft/s.  One minutes later, another girl

starts at a point A and runs north at a rate of 8 ft/s.  At what rate is the distance between them

changing one minute after the second girl starts?

Variables:

x = distance from A for first girl.
y = distance from A for second girl.
z = distance between the two.

Rates:

dx/dt = speed of first girl,  GIVEN as  10 ft/sec
dy/dt = speed of second girl,  GIVEN as  8 ft/sec
dz/dt = rate of increase of distance,  TO BE FOUND.

Relation:  There is a right triangle here, and  x^2 + y^2 = z^2.

Differentiate:

2x dx/dt + 2y dy/dt = 2z dz/dt

x dx/dt + y dy/dt = z dz/dt

Substitute.  We have rates but:

Difficulty encountered:  We don't have current values for  x,y,z.
Resolution of this issue:  Use  t = 60 and the relation to find them.

x(60) = 60*10 = 600 feet.
y(60) = 60(8) = 480 feet.

x^2 + y^2 = z^2

600^2 + 480^2 = z^2

z = sqrt(600^2 + 480^2) = sqrt(120^2(5^2 + 4^2)) = 120 sqrt(41)

Ready to go.

x dx/dt + y dy/dt = z dz/dt

600(10) + 480(8) = 120 sqrt(41) dz/dt

5(10) + 4(8) = sqrt(41) dz/dt

50 + 32 = sqrt(41) dz/dt
         82
dz/dt = --------, which you can simplify a little.
       sqrt(41)

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I think you now know enough to do 3,4 yourself.  If you still have trouble, send me another query, but INCLUDE what you already managed to do on them -- don't just repeat the problems.  Try to organize the problem as I outlined above, and IN PARTICULAR,

State very clearly what the variables stand for.  In the second problem, if you wrote:

x = first girl

you would get nowhere.  The most important part of any verbal problem is the statement of the variables.  Do a good job on this and the problem flows.  Get lazy on this part and you get stuck.

-----Answer-----
Questioner:  Rachel
Category:  Calculus
 
Subject:  Related Rates
Question:  Thank you so much for your help!!! I feel like I'm beginning to really grasp the concept.  I do have a couple questions about number 3:
3)As sand leaks out of a hole in a container, it forms a conical pile whose height is always equal to the radius of the pile.  If the height of the pile is increasing at a rate of 6 inches per minute, find the rate at which the sand is leaking out of the container when the height of the pile is 10 inches.

I think the formala that will relate all the variable together is V = pi/3 * r^2 * h, but i'm not positive if the rate of change of the volume of the pile is equal to the rate at which the sand is leaking out of the container.

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That would seem logical.  If the volume of the pile is increasing, that means it's getting more sand.  Where did you think that sand is coming from?

Nevertheless, the conditions of this problem are a bit strange, because normally the sand leaks out at a constant rate and the height of the pile increases at a varying rate -- here it seems the other way around.  No matter, though; let's do our standard stuff.

Variables:

h = the height of the conical pile of sand.
r = the radius of the conical pile, WHICH IS ALWAYS EQUAL TO h, says the problem.
V = the volume of the conical pile.

Rates:

dh/dt = the rate of increase of the height, GIVEN as  6 in/min.
dr/dt, which is irrelevant, r = h.
dV/dt = the rate of increase of volume, TO BE FOUND.

Relation:

V = 1/3 pi r^2 h = 1/3 pi r^3,  since  r = h.

Differentiate:

dV/dt = pi r^2 dr/dt

Substitute:

dr/dt = 6,  h = 10

dV/dt = pi (10)^2 (6) = 600 pi  cubic inches/min
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4)A model rocket enthusiast launches a rocket vertically from a point that is 5 miles away from his tracking device.  Assume the launch site and the tracking device are at the same elevation.  

For the first 20 seconds of flight, theta, the angle of elevation, changes at a constant rate of 2 degrees/s.  Find the velocity of the rocket in miles per hour when the angle of elevation is 30 degrees.

For this, a diagram is a good idea.  Put the observer at O, the launch point at L, and the position of the rocket at R, with  OLR being a right triangle, and theta = angle ROL.

Variables:  (I will write T for theta in any expressions.)

T = angle ROL
y = the height LR of the rocket.

Rates:

dT/dt = the rate of increase of theta, GIVEN as 2 degrees/sec.
dy/dt = the velocity of the rocket, TO BE FOUND.

Now I see some obvious issues here.  ("Issues" is our 21st-century word for problems.)

dT/dt is in degrees/second, but dy/dt will be in miles per hour.  So we have to use a common unit here.  I'll do everything in things/hour.  

T should never, never, never, be in degrees -- it should be in radians.  2 degrees = pi/90 radians.

[I'm sure your teacher has already said "No, no, ze degrees, zey are for children, ve alvays use ze radians in ze calculus."]

Putting those together, we have:

dT/dt = pi/90 * 3600 degrees/hour. = 40 pi deg/hour

Relation:

tan T = y/5,  from the right triangle.

Differentiate:

sec^2(T) dT/dt = 1/5 dy/dt

Substitute:  T = 30 degrees,  dT/dt = 40 pi degrees/hour.

sec(30) = 1/cos 30 = 1/(sqrt(3)/2)) = 2/sqrt(3),
sec^2(30) = 4/3

Finally:

4/3 (40 pi) = 1/5 dy/dt
160pi/3 = 1/5 dy/dt

dy/dt = 800 pi/3 miles per hour, which is about 850 or so, I guess.


Answer
Questioner:  Rachel
Category:  Calculus
Subject:  Related Rates
Question:  how would i differentiate (1/sinY)(ysin(theta))?!?! i know it involves the chain rule, but i'm unsure how to carry that out in this problem
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Hi, again, Rachel,

I'm not sure what you're asking here.   

In your expression: (1/sinY)(ysin(theta))

What is the independent variable?  Are you assuming that y and theta are functions of a third variable?  When you say 'differentiate', what are you differentiating with respect to?

In all the R-R problems, you were differentiating implicitly with respect to t.  What's going on here?

If this is part of some problem, you should send along the entire problem, along with anything you already did.  I don't see it as having anything to do with the four problems you were sending before.