Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius 2.
A little investigation will lead you to the formula that if a hole of radius r is drilled through a sphere of radius R, the remaining volume is just 4pi/3 (R^2 - r^2)^(3/2)
In this case, that would be 4pi/3 * 3√3 = 4√3 pi
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You might also look up the napkin ring problem, where it is shown that if a hole of length h is drilled through a sphere, the remaining volume is independent of the radius of the sphere!
In this case, h/2 = R^2 - r^2 = √3, so h = 2√3
The remaining volume is pi/6 h^3 = pi/6 * 24√3 = 4√3 pi
thanks!
To find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius 2, we can subtract the volume of the hole from the volume of the sphere.
Step 1: Calculate the volume of the sphere.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
In this case, the radius of the sphere is 2, so the volume of the sphere is:
V_sphere = (4/3)π(2^3) = (4/3)π(8) = 32/3π.
Step 2: Calculate the volume of the hole.
The volume of the hole is a cylinder with a radius of 1 and a height equal to the diameter of the sphere.
The diameter of the sphere is twice the radius, so it is 2 * 2 = 4.
The height of the cylinder is equal to the diameter of the sphere, so h = 4.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
In this case, the radius of the hole is 1 and the height is 4, so the volume of the hole is:
V_hole = π(1^2)(4) = 4π.
Step 3: Calculate the volume that remains.
The volume that remains after the hole is bored through the sphere is the difference between the volume of the sphere and the volume of the hole.
V_remain = V_sphere - V_hole.
Substituting the values we calculated earlier, we have:
V_remain = (32/3π) - (4π) = 32/3π - 12/3π = (32 - 12)/3π = 20/3π.
Therefore, the volume that remains after the hole is bored through the center of the solid sphere is 20/3π.
To find the volume that remains after a hole is bored through the center of a solid sphere, we need to subtract the volume of the cylindrical hole from the volume of the sphere.
The volume of a sphere is given by the formula:
V_sphere = (4/3) * π * r^3
Given that the radius of the sphere is 2, we can calculate its volume:
V_sphere = (4/3) * π * 2^3
= (4/3) * π * 8
= 32/3 * π
Now, let's find the volume of the cylindrical hole. The radius of the hole is given as 1, and its height is equal to the diameter of the sphere, which is 2 times the radius of the sphere.
The volume of a cylinder is given by the formula:
V_cylinder = π * r^2 * h
Since the cylinder is centered and passes through the center of the sphere, its height is equal to the diameter of the sphere:
h = 2 * 2
= 4
Now we can calculate the volume of the cylindrical hole:
V_cylinder = π * 1^2 * 4
= 4π
Finally, to find the remaining volume, we subtract the volume of the cylinder from the volume of the sphere:
V_remaining = V_sphere - V_cylinder
= (32/3 * π) - (4π)
= (32/3 - 4) * π
= (32π/3 - 12π/3)
= (20π/3)
Therefore, the volume that remains after the hole is bored through the center of the solid sphere is (20/3)π.