Rate of change

[Specter]

I was looking through an old Calculus textbook and came upon this R.O.C. problem. I don't think it's too difficult, but I really have no idea how to solve it.

quote:

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The radius (r) of a sphere is increasing at a constant rate 0.04 cm/sec.

Find:
a. At the time when the radius is 10 centimeters, what is the rate of increase of its volume?
b. At the time when the volume is 36pi cm³, what is the rate of increase of the area of a cross section through the center of the sphere?
c. At the time when the volume and the radius are increasing at the same numerical rate, what is the radius?

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I think I know how to define some of the changing variables as derivatives.
Based on what I remember, I think I can say...
dr/dt = 0.04
a. Find dr/dt when r=10 cm.
b. Find dA/dt when V=36 pi.
c. When dV/dt = dr/dt, find r.
(right?)

V_sphere=4/3*pi*r³
A_circle=pi*r²

That is about as far into Calculus that I can remember.

[Dr. Borodog]

dV/dt = 4pr² dr/dt
dA/dt = 2pr dr/dt

That should get you started.


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You will respect my philosophai.

[Dr. Borodog]

Although, dV/dt can never equal dr/dt, as they are dimensionally different, so I don't know what they want there.

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You will respect my philosophai.

[mith]

It's asking where the numbers are the same, ignoring the dimensions.

[Dr. Borodog]

Thanks mith. I should have gotten that from the original statement: ". . . increasing at the same numerical rate . . ."

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You will respect my philosophai.

[This message has been edited by Dr. Borodog (edited 03-25-2004 09:41 PM).]

[Samadhi]