Optimization calculus 1

Question

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?

Answer 1

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).