Final answer:
To find the diameter of the hole drilled through the solid, we calculate the volume of the original solid of revolution, take one-fourth of that volume, and equate it to the volume formula of a cylinder with the diameter being the variable to solve for.
Step-by-step explanation:
The problem involves calculating the volume of a solid of revolution around the y-axis and then finding the diameter of a hole drilled through the center such that one-fourth of the volume is removed. Given the equation for the bounded region is y = 1/8x2, we first find the volume V of the solid by using the method of disks or washers and revolving the curve around the y-axis. Recognizing that the volume of the drilled hole represents one-fourth of the original volume, we express this as Vhole = V/4. Then we use the formula for the volume of a cylinder V = πr2h to relate the volume of the hole to its radius (and hence diameter), since the height h would be the same for the full solid and the drilled hole.
To find the diameter D of the hole, we solve the equation Vhole = π(D/2)2h. Simplifying this expression and solving for D would give us the required diameter to one-fourth of the original volume.