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January 30th, 2007, 02:55 AM   #1
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Cylinder in a Sphere

Inscribed a right circular cylinder of height h and radius r in a sphere of fixed radius R. Express the volume V of the cylinder as a function of h.

HINTS GIVEN:

(1) V = pi(r^2)(h).

(2) R = the hypotenuse and r = one of the legs of the right triangle in the cylinder.
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January 30th, 2007, 03:13 AM   #2
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I don't get the point of this problem. It looks like a complicated trick question to me.

Quote:
Inscribed a right circular cylinder of height h and radius r in a sphere of fixed radius R. Express the volume V of the cylinder as a function of h.
Well, it doesn't matter where the cylinder is, how it is oriented, whether it is inscribed, or whatever.
A cylinder's volume is always related to its height by the equation:
V=pi(r^2)(h), so they actually already gave you the answer to the question in their hints.
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January 30th, 2007, 03:20 AM   #3
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Oh, OK, I see now what they're getting at. They want to get rid of the 'r' variable and express the whole thing in term of h only. So, all we must do is find an expression that relates r and h.
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January 30th, 2007, 01:09 PM   #4
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ok

How do we find an expression that relates r and h?
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January 30th, 2007, 02:31 PM   #5
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Follow the second hint - the unmentioned third side of the triangle is of length h/2. Pythagoras's theorem gives the relationship needed.
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February 2nd, 2007, 03:52 PM   #6
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ok

Can you show me how?
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February 2nd, 2007, 05:14 PM   #7
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Sketch the situation. You should find it easy to spot a triangle containing a right angle, with sides are of lengths r, h/2 and R (for the hypotenuse).
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February 3rd, 2007, 08:35 AM   #8
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ok

We have:

r, h/2 and R = hypotenuse, right?

I get that part but what am I solving for?

Do I use the Pythagorean Theorem to solve for r, h/2 or R?
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February 3rd, 2007, 12:03 PM   #9
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In order, "yes", "r" and "yes, r".
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February 3rd, 2007, 01:45 PM   #10
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ok

You are saying to solve for r, h/2 and R INDIVIDUALLY IN ORDER?
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