A solid cone of height 12cm and radius 9 cm is recast into a solid sphere. Calculate the surface area of the sphere.
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cone: v = 1/3 pi r^2 h
using that v, then for the sphere, find its radius and then area:
v = 4/3 pi r^3
a = 4 pi r^2
To calculate the surface area of the sphere, we need to determine the radius of the sphere first.
Given that the cone has a height of 12 cm and a radius of 9 cm, we can use the formula for the volume of a cone to find the volume of the cone.
The volume of a cone is given by V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.
Substituting the given values, we have:
V = (1/3)π(9^2)(12)
V = (1/3)π(81)(12)
V = (1/3)π(972)
V = 324π cm^3
Since the cone is recast into a solid sphere, the volume of the sphere will be equal to the volume of the cone.
The volume of a sphere is given by V = (4/3)πr^3, where V is the volume and r is the radius.
Substituting the volume of the cone, we have:
324π = (4/3)π(r^3)
(4/3)π(r^3) = 324π
r^3 = (324π * 3) / 4
r^3 = 243π
r = ∛(243π)
r = 9 cm
Now that we have determined that the radius of the sphere is 9 cm, we can calculate the surface area of the sphere.
The surface area of a sphere is given by A = 4πr^2, where A is the surface area and r is the radius.
Substituting the radius, we have:
A = 4π(9^2)
A = 4π(81)
A = 324π
Therefore, the surface area of the sphere is 324π cm^2.
To calculate the surface area of the sphere, we need to find the radius of the sphere first.
Since the cone is recast into a sphere, the volume of the cone will be equal to the volume of the sphere. Therefore, we can set up the equation:
(1/3) * pi * (r_cone)^2 * h_cone = (4/3) * pi * (r_sphere)^3
Substituting the given values:
(1/3) * pi * (9 cm)^2 * 12 cm = (4/3) * pi * (r_sphere)^3
Simplifying:
324 pi = (4/3) * pi * (r_sphere)^3
Dividing both sides by (4/3) * pi, the pi terms cancel out:
r_sphere^3 = (324 * (3/4)) cm^3
r_sphere^3 = 243 cm^3
Taking the cube root of both sides gives us:
r_sphere = 6 cm
Now that we have the radius of the sphere, we can calculate the surface area using the formula:
Surface area = 4 * pi * r_sphere^2
Substituting the value of the radius:
Surface area = 4 * pi * (6 cm)^2
Surface area = 4 * pi * 36 cm^2
Surface area = 144 pi cm^2
So, the surface area of the sphere is 144 pi cm^2.