the volume of a cone and a cylinder is in the ratio 2:3. the height and diameter of the cone are 18cm and 12cm respectively. the diameter of the cylinder is 30cm. find the height of the cylinder (π=22/7)
To find the height of the cylinder, we first need to find the volume of both the cone and the cylinder.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.
Given that the height of the cone is 18cm and the diameter is 12cm, we can find the radius of the cone using the formula r = d/2 = 12/2 = 6cm.
Thus, the volume of the cone is V_cone = (1/3)(22/7)(6^2)(18) = 792cm^3.
Let x be the height of the cylinder.
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius of the base and h is the height of the cylinder.
Given that the diameter of the cylinder is 30cm, we can find the radius of the cylinder using the formula r = d/2 = 30/2 = 15cm.
Since the volume of the cone is 2/3 times the volume of the cylinder, we have (792cm^3)/(V_cylinder) = (2/3).
This can be rewritten as V_cylinder = (3/2)(792cm^3) = 1188cm^3.
Substituting the known values into the volume formula for the cylinder, we get (22/7)(15^2)(x) = 1188cm^3.
Simplifying, (22/7)(225)(x) = 1188cm^3.
Multiplying both sides by (7/22)(1/225), we get x = (1188cm^3)(7/22)(1/225).
Simplifying, x = (11/5)cm.
Therefore, the height of the cylinder is 11/5 cm or 2.2 cm.