A ball of radius 10 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid.
To find the volume of the resulting solid, we can subtract the volume of the hole from the volume of the original ball.
The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere.
Let's start by finding the volume of the original ball. Since the radius of the ball is 10, the volume is given by V1 = (4/3)π(10^3) = (4/3)π(1000) = (4/3)π(1000) = 4000π.
Next, we calculate the volume of the hole. The hole has a radius of 8, so the volume is given by V2 = (4/3)π(8^3) = (4/3)π(512) = 682.67π (approx.).
Finally, we subtract the volume of the hole from the volume of the original ball to find the volume of the resulting solid.
Volid = V1 - V2 = 4000π - 682.67π = 3317.33π (approx.).
Therefore, the volume of the resulting solid is approximately 3317.33π.
To find the volume of the resulting solid, we can subtract the volume of the drilled hole from the volume of the ball.
1. Calculate the volume of the ball:
The volume of a sphere is given by the formula: V = (4/3)πr^3, where r is the radius.
Substituting the given radius of the ball, which is 10, into the formula, we have:
V_ball = (4/3)π(10^3) = (4/3)π(1000) = (4/3)π(1000) = 4000π.
2. Calculate the volume of the hole:
The volume of a cylinder is given by the formula: V = πr^2h, where r is the radius and h is the height.
The hole drilled through the center of the ball forms a cylinder.
The radius of the hole is given as 8, which means the radius of the cylinder is also 8.
The height of the cylinder can be determined by considering that it passes through the center of the ball, so the height will be twice the radius of the ball, which is 2 x 10 = 20.
Substituting the values into the formula, we have:
V_hole = π(8^2)(20) = π(64)(20) = 1280π.
3. Calculate the volume of the resulting solid:
To find the volume of the resulting solid, we need to subtract the volume of the hole from the volume of the ball:
V_result = V_ball - V_hole = 4000π - 1280π = 2720π.
Therefore, the volume of the resulting solid is 2720π cubic units.
whole ball = 4/3(pi)(10^3)= (4000/3)(pi)
now drill out the integral of Pi(y^2)dx from -4 to 4 or
2pi(integral)(100-x^2)dx from 0 to 4
= 2pi [ 100x - (1/3)x^3] from 0 to 4
= 2pi(400 - 64/3]
= 2pi(1136/3)
so the volume of the solid is
4000pi/3 - 2275pi/3 = 1728pi/3
please check my arithmetic.