Difference between revisions of "1992 AJHSME Problems/Problem 16"
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<math>\text{(E)}\ \text{None of the above}</math> | <math>\text{(E)}\ \text{None of the above}</math> | ||
| + | |||
| + | ==solution== | ||
| + | |||
| + | |||
| + | (B) Cylinder can be obtained b stacking one copy | ||
| + | of the given cylinder on top of another. The | ||
| + | formula for the volume of a cylinder with radius | ||
| + | r and the h is V= pi r^2h. Use this to show that | ||
| + | none of the other cylinders has twice the volume | ||
| + | of the given cylinder: | ||
| + | |||
| + | Cylinder- given: pi x 10^2 x 5= 500pi | ||
| + | (A): pi | ||
Revision as of 18:12, 4 November 2012
Problem
![[asy] draw(ellipse((0,-5),10,3)); fill((-10,-5)--(10,-5)--(10,5)--(-10,5)--cycle,white); draw(ellipse((0,0),10,3)); draw((10,0)--(10,-5)); draw((-10,0)--(-10,-5)); draw((0,0)--(7,-3*sqrt(51)/10)); label("10",(7/2,-3*sqrt(51)/20),NE); label("5",(-10,-3),E); [/asy]](http://latex.artofproblemsolving.com/4/8/a/48a018e84e421aaed03f4192d4e70246b227fec7.png)
Which cylinder has twice the volume of the cylinder shown above?
![[asy] unitsize(4); draw(ellipse((0,-5),20,6)); fill((-20,-5)--(20,-5)--(20,5)--(-20,5)--cycle,white); draw(ellipse((0,0),20,6)); draw((20,0)--(20,-5)); draw((-20,0)--(-20,-5)); draw((0,0)--(14,-3*sqrt(51)/5)); label("20",(7,-3*sqrt(51)/10),NE); label("5",(-20,-4),E); label("(A)",(0,6),N); draw(ellipse((31,-7),10,3)); fill((21,-7)--(41,-7)--(41,7)--(21,7)--cycle,white); draw(ellipse((31,3),10,3)); draw((41,3)--(41,-7)); draw((21,3)--(21,-7)); draw((31,3)--(38,3-3*sqrt(51)/10)); label("10",(34.5,3-3*sqrt(51)/20),NE); label("10",(21,-4),E); label("(B)",(31,6),N); draw(ellipse((47,-15.5),5,3/2)); fill((42,-15.5)--(42,-15.5)--(42,15.5)--(42,15.5)--cycle,white); draw(ellipse((47,4.5),5,3/2)); draw((42,4.5)--(42,-15.5)); draw((52,4.5)--(52,-15.5)); draw((47,4.5)--(50.5,4.5-3*sqrt(51)/20)); label("5",(48.75,4.5-3*sqrt(51)/40),NE); label("10",(42,-6),E); label("(C)",(47,6),N); draw(ellipse((73,-10),20,6)); fill((53,-10)--(93,-10)--(93,5)--(53,5)--cycle,white); draw(ellipse((73,0),20,6)); draw((53,0)--(53,-10)); draw((93,0)--(93,-10)); draw((73,0)--(87,-3*sqrt(51)/5)); label("20",(80,-3*sqrt(51)/10),NE); label("10",(53,-6),E); label("(D)",(73,6),N); [/asy]](http://latex.artofproblemsolving.com/1/3/4/1349b963179bfa1b7e83dca7b965388f5d7832b1.png)
solution
(B) Cylinder can be obtained b stacking one copy of the given cylinder on top of another. The formula for the volume of a cylinder with radius r and the h is V= pi r^2h. Use this to show that none of the other cylinders has twice the volume of the given cylinder:
Cylinder- given: pi x 10^2 x 5= 500pi (A): pi
