<p>can anyone solve this using single variable calc</p>
<p>What is the largest volume right circular cylinder that can be inscribed in a sphere with radius r?</p>
<p>i know that what I have to do is get h in terms of r
volume of a cylinder is pi r^2 h</p>
<p>thanks</p>
<p>Hi,</p>
<p>The height of the cylinder can be related to the radius of the cylinder and sphere by using the Pythagoras theorem. The hypotenuse of the right angle triangle is the diameter of the circle. Therefore, h^2=4r^2+4R^2, where R is the radius of the sphere, and r is the radius of the cylinder. If you substitute h in the formula for the volume of the cylinder and differentiate with respect to r, and set it to zero, you will obtain r = sqrt(2/3)R. The max volume of the cylinder turns out to be 1/(sqrt(3)) of the volume of the sphere.</p>
<p>I hope this helps.
mdabral</p>
<p>mdabral:
Should that formula be (h/2)^2 = R^2 - r^2, from which h^2 = 4(R^2 - r^2)?</p>
<p>Sorry, I wrote the equation wrong. You are right, the diameter of the sphere is the hypotenuse. </p>
<p>mdabral</p>
<p>thanks a lot</p>
<p>im not understanding how you go from r = rad(2/3 R) to the 1/rad3 times the volume of the sphere?</p>
<p>I think you misread mdabral's post, he/she had r = R sqrt(2/3)</p>
<p>Vol(cylinder) = pi r^2 h
= pi (2/3) R^2 (2sqrt(R^2 - r^2))
= pi (4/3) R^2 (sqrt(R^2 - (2/3)R^2))
= pi (4/3) sqrt(1/3) R^3</p>
<p>Vol(sphere) = (4/3) pi R^3</p>
<p>Vol(cylinder) / Vol(sphere) = sqrt(1/3)</p>