A metal cylinder with a 4 inch radius and a 10 inch altitude is melted and recast into solid right circular cones each with a 1 inch radius and 2 inch altutudes. What is the number of cones formed?
Dig out your formulas for volume of cylinder and cone:
cylinder V=Bh = pi * r^2 * h = 160pi
cone: V = 1/3Bh = 1/3 pi r^2 * h = 2/3 pi
SO, 160/(2/3) = 240 cones.
To find the number of cones formed, we need to determine the volume of the metal cylinder and the volume of each cone, and then divide the volume of the cylinder by the volume of each cone.
First, let's find the volume of the metal cylinder using the formula for the volume of a cylinder: V = πr^2h, where V is the volume, r is the radius, and h is the altitude.
Given:
Radius of the metal cylinder (r) = 4 inches
Altitude of the metal cylinder (h) = 10 inches
Volume of the metal cylinder (V) = π(4^2)(10) = 160π cubic inches
Next, let's find the volume of each cone using the formula for the volume of a cone: V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the altitude.
Given:
Radius of each cone (r) = 1 inch
Altitude of each cone (h) = 2 inches
Volume of each cone (V) = (1/3)π(1^2)(2) = (2/3)π cubic inches
Finally, to find the number of cones formed, we will divide the volume of the metal cylinder by the volume of each cone:
Number of cones = (Volume of metal cylinder)/(Volume of each cone)
= (160π)/((2/3)π)
= (160π)(3/2π)
= 240
Therefore, the number of cones formed is 240.