Optimization: Inscribed Shapes

[Oneiromancy]

I don't understand the general procedure to approach these "inscribed shape" optimization problems. I know that in general for optimization you get your objective function (the thing you want to max/min), your constraint, find the domain, then do 1st and 2nd derivative tests and basically plug in numbers after that.

Here's an example:

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.

 

[stapel]

I don't understand the general procedure to approach these "inscribed shape" optimization problems....

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.

Please reply showing what you have tried and specifying at what point you are getting stuck. You wrote down the formulas for the volumes of the two shapes, you solve the sphere-volume formula for the radius r, and... then what?

Please be complete. Thank you!

Eliz.

 

[Oneiromancy]

Ok well, I assume this is going to involve similar triangles somehow since in my book it shows a simple picture showing a cone labeled with a right angle from the "point" down to the base inside an unlabeled sphere.

What I don't get it how the sphere is going to restrain the cone's shape. This would be a pretty easy problem if it just involved a cone on its own.

 

[skeeter]

start by making a sketch ...

draw a circle of radius 3 centered at the origin.
inscribe an isosceles triangle inside the circle such that its base is parallel and above the x-axis, and its vertex angle is at the point (0,-3).

when the circle and triangle are rotated about the y-axis, a cone inscribed in a sphere is formed.

let the base of the triangle have length = 2x ... cone's radius = x
height of the triangle = height of the cone = y + 3

relationship between x and y ... \(\displaystyle y = \sqrt{9 - x^2}\)

volume of a cone is
\(\displaystyle V = \frac{\pi}{3}r^2 h\)

for the cone inscribed in the sphere of radius 3 ...

\(\displaystyle V = \frac{\pi}{3} x^2 \left[\sqrt{9 - x^2} + 3 \right]\)

now find dV/dx and maximize V.

 

[Oneiromancy]

Where did you get y = sqrt (9 - x^2) ?

 

[skeeter]

what's the equation of a semicircle of radius 3 ?