Question 1012281
cylinder height = 7 cm and volume - 112 * pi cubic cm.


henry fills the cup with water all the way to the top.


he then pours all the water into a cone.


presumably this also fills up the cone.


the cone has double the radius of the cylinder.


what is the height?


the volume of the cone must be the same as the volume of the cylinder.


the formula for the volume of a cone is v = 1/3 * pi * r^2 * h


the radius of the cone is twice the radius of the cylinder.


the radius of the cylinder is is not provided so we have to solve for it.


the formula for the volume of a cylinder is pi * r^2 * h.


since the volume is 112, the formula becomes 112 = pi * r^2 * 7, because the height is given as 7 cm.


we can solve for r^2 to get r^2 = 112 / (pi * 7) which becomes r^2 = 16 / pi.


solve for r to get r = 4 * sqrt(pi).


since the radius of the cone is twice the radius of the cylinder, then the radius of the cone is 8 * sqrt(pi).


we get formula for the volume of the cone becomes 112 = 1/3 * pi * (8 * sqrt(pi))^2 * h


solve for h to get h = 112 / (1/3 * (8 * sqrt(pi))^2)


simplify to get h = (112 * 3) / (64 * pi) which becomes h = 336 / (64 * pi) which becomes h = 5.25 cm.


so, we have:


r for the cylinder = 4/sqrt(pi)
h for the cylinder = 7


r for the cone = 8/sqrt(pi)
h for the cylinder = 5.25


volume for each = 112.


formula for volume of a cylinder is v = pi * r^2 * h.
v = 112
r = 4/sqrt(pi), r^2 = 16/pi
h = 7
formula becomes 112 = pi * 16/pi * 7 = 16 * 7 = 112.
this checks out ok.


formula for volume of a cone is v = 1/3 * pi * r^2 * h.
v = 112
r = 8/sqrt(pi), r^2 = 64/pi
h = 5.25
formula becomes 112 = 1/3 * pi * 64/pi * 5.25 = 1/3 * 64 * 5.25 = 112.
this checks out ok as well.


your solution is that the height of the cone is 5.25 centimeters.