How do related rates apply to circles, rectangles, and spheres?

[courtrigrad]

Hello all

Just had questions on related rates:

1. The radius of a circle is growing by $$\frac{dr}{dt} = 7$$. How fast is the circumference growing? Ok so $$C = 2\pi r$$ and $$\frac{dC}{dr} = 2\pi \frac{dr}{dt} = 2\pi(7) = 14\pi$$

2. #1 has some amazing implications. Suppose you want to put a rope around the Earth that any 7-footer can walk under. If the distance is 24,000 miles, what is the additional length of rope? Do I just put $$C = 24,000$$? I am not sure if I understand what it is asking.

3. The sides of a rectangle increase in such a way that $$\frac{dz}{dt} = 3\frac{dy}{dt}$$ where z is the diagonal. At the instant when $$x = 4 y = 3$$ what is the value of $$\frac{dx}{dt}$$? So $$x^2 + y^2 = z^2$$. $$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2$$ So then do I just substitute in the given values do get $$\frac{dx}{dt}$$? How would I use the fact that $$\frac{dx}{dt} = 3\frac{dy}{dt}$$?

4. Air is being pumped into a spherical balloon at the rate of $$5.5$$ cubic inches per minute. Find the rate of change of the radius when the radius is 4 inches. Ok so I know that $$V = \frac{4}{3}\pi r^3$$. So $$\frac{dV}{dt} = 5.5$$ So $$5.5 = 4\pi (4)^{2} \frac{dr}{dt}$$. I get $$\frac{5.5}{64\pi}$$ Is this correct?


Thanks

 

[learningphysics]

First question is right. I don't understand the second question. Have you copied it word for word?

For the third question the equation should be
$$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}$$

Your question is confusing... Is:
$$\frac{dx}{dt} = \frac{dz}{dt} = 3\frac{dy}{dt}$$

In one part you've written dx/dt=3dy/dt... another part you've written dz/dt=3dy/dt.

 

[dextercioby]

Your last question is solved correctly,though you should have added the unit...

As for the second,the way i see it...You don't need too much data...

Daniel.

 

[courtrigrad]

I copied it word for word. I am sorry. For #3 it should be: $$\frac{dz}{dt} = 3\frac{dy}{dt}$$

 

[learningphysics]

I copied it word for word. I am sorry. For #3 it should be: $$\frac{dz}{dt} = 3\frac{dy}{dt}$$

That's ok.

For the second question, I guess that it's asking for the length of rope a 7ft can walk under around Earth - circumference of earth.

For the third question... I believe there's insufficient data to get an exact value for dx/dt. Double check the question, to see if there's more data.

 

[courtrigrad]

for #3 we have The sides of a rectangle increase such that $$\frac{dz}{dt} = 1$$ and $$\frac{dx}{dt} = 3\frac{dy}{dt}$$. Find $$\frac{dx}{dt}$$ when $$x = 4, \ y = 3$$

I get: $$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}$$. Substituting $$x,y,z$$ we get $$8\frac{dx}{dt} + 6\frac{dy}{dt} = 10$$ Substituting in $$\frac{dx}{dt} = 3\frac{dy}{dt}$$ I got $$\frac{dx}{dt} = 1$$ Is this correct?


Whoops

 

[learningphysics]

I get: $$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}$$. Substituting $$x,y,z$$ we get $$8\frac{dx}{dt} + 6\frac{dy}{dt} = 10$$ Substituting in $$\frac{dx}{dt} = 3\frac{dy}{dt}$$ I got $$\frac{dx}{dt} = 1$$ Is this correct?


Whoops

Looks good.