Question 1198554
Let the radius of the cone be {{{r[1]}}}, the radius of the cylinder be {{{r[2]}}}, the height of the cone {{{h[1]}}}, and the height of the cylinder {{{h[2]}}}. Then, we have the equation {{{(1/3)*pi*r[1]^2*h[1]=pi*r[2]^2*h[2]}}}. Our goal is to find {{{h[1]/h[2]}}}. First, we can divide {{{pi}}} from both sides to get {{{(1/3)*r[1]^2*h[1]=r[2]^2*h[2]}}}. We can multiply both sides by 3, then divide both sides by {{{r[1]^2}}} to get {{{h[1]=3*(r[2]^2*h[2])/r[1]^2}}}. Dividing both sides by {{{h[2]}}}, we get {{{h[1]/h[2]=3*r[2]^2/r[1]^2}}}
Since we know that the base of the cone is two-thirds the area of the base of the cylinder, we have the equation {{{pi*r[1]^2=(2/3) pi*r[2]^2}}}. Dividing both sides by {{{pi}}}, we get {{{r[1]^2=(2/3)*r[2]^2}}}. Multiplying both sides by {{{3/2}}}, we get {{{(3/2)*r[1]^2=r[2]^2}}}. Finally, dividing both sides by {{{r[1]^2}}} gives us {{{r[2]^2/r[1]^2=3/2}}}. We can plug this into the first equation to get {{{h[1]/h[2]=3*3/2=highlight(9/2)}}}. 
The question doesn't state whether you want the ratio of the height of the cone over the height of the cylinder (which was what I derived), or the ratio of the height of the cylinder over the height of the cone. If you want that, the answer would instead be {{{2/9}}}.