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Optimization calculus 1

A cylinder is inscribed in a right circular cone of height \displaystyle 11 inches and radius (at the base) equal to \displaystyle 6 inches. What are the dimensions of such a cylinder that has maximum volume?
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1 Answer

edited version of your question:  A cylinder is inscribed in a right circular cone of height 11 inches and radius (at the base) equal to  6 inches. What are the dimensions of such a cylinder that has maximum volume?
 
Draw a side view of a slice through the cone and cylinder, showing a rt triangle with h=11,r=6.
The slice through the cylinder shows a rectangle  of base x and height y.
By similar triangles, 11-y : x = y : 6-x and thus you can express y in terms of x. This will be understood if you draw the figure.
 
Now write the volume of the cylinder: V = pi (x2)(y) and then substitute for y. 
Now use calculus techniques to maximize V(x).