Final answer:
To calculate the volume of the resulting solid when a hole is drilled through a sphere, find the volume of the original sphere and then subtract the volume of the cylindrical hole. Use the formulas for the volume of a sphere and the volume of a cylinder to perform this calculation.
Step-by-step explanation:
To find the volume of the resulting solid when a ball of radius 15 has a round hole of radius 6 drilled through its center, we must consider the volume of the sphere and subtract the volume of the cylindrical hole.
Volume of the sphere:
The formula for the volume of a sphere is V = (4/3)πr³. For a sphere of radius 15, this becomes V = (4/3)π(15³). We calculate this volume first.
Volume of the cylindrical hole:
The volume of a cylinder is given by the formula V = πr²h. Here, h is the height of the cylinder, which is equal to the diameter of the sphere (2 * 15) because the hole is drilled through the center, and r is the radius of the hole, which is 6. We calculate this volume separately.
Final volume of the solid:
The final volume of the solid is the volume of the sphere minus the volume of the cylindrical hole. Using the formulas and values given, we can compute the final volume.