A solid metal cylinder with a 4-in. radius and a 10-in. altitude is melted and recast into solid right circular cones each with a 1-in. radius and a 2-in. altitude. The number of cones formed is
vol of cyl = πr^2h = 160π
vol of cone = π/3r^2h = 2π/3
160π / (2π/3) = 240 cones
To find the number of cones formed, we need to calculate the volume of the solid metal cylinder and the volume of each cone, and then divide the volume of the cylinder by the volume of each cone.
The volume of a cylinder is given by the formula:
V_cylinder = π * r^2 * h
where "r" is the radius of the cylinder and "h" is the altitude (or height) of the cylinder.
The volume of a cone is given by the formula:
V_cone = (1/3) * π * r^2 * h
where "r" is the radius of the cone and "h" is the height of the cone.
First, let's calculate the volume of the metal cylinder:
V_cylinder = π * (4 in)^2 * 10 in
V_cylinder = 160π in^3
Now, let's calculate the volume of each cone:
V_cone = (1/3) * π * (1 in)^2 * 2 in
V_cone = (1/3) * 1π in^3
V_cone = (1/3)π in^3
To find the number of cones formed, we need to divide the volume of the cylinder by the volume of each cone:
Number of cones = V_cylinder / V_cone
Number of cones = (160π in^3) / ((1/3)π in^3)
Number of cones = (160*3) / 1
Number of cones = 480 cones
Therefore, the number of cones formed is 480.
To find the number of cones formed, we need to compare the volume of the original cylinder with the volume of each cone.
First, let's find the volume of the original cylinder. The formula for the volume of a cylinder is given by V = πr^2h, where r is the radius and h is the altitude.
V_cylinder = π * (4 in.)^2 * (10 in.)
= π * 16 in.^2 * 10 in.
= 160π in.^3
Next, let's find the volume of each cone. The formula for the volume of a cone is given by V = (1/3)πr^2h, where r is the radius and h is the altitude.
V_cone = (1/3) * π * (1 in.)^2 * (2 in.)
= (1/3) * π * 1 in.^2 * 2 in.
= (2/3)π in.^3
Now, we can find the number of cones formed by dividing the volume of the original cylinder by the volume of each cone.
Number of cones formed = V_cylinder / V_cone
= (160π in.^3) / [(2/3)π in.^3]
= (160 in.^3) / (2/3) in.^3
= (160 in.^3) * (3/2) in.^3
= 240 in.^3 / in.^3
= 240
Therefore, the number of cones formed is 240.