Question

Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations z = 25e^-(x²+y²)/4, z = 0, and x² + y² = 16 when one-tenth of the volume of the solid is removed.

Answer 1

To determine the diameter of the hole, integrate the equation z = 25e^-(x²+y²)/4 over the given region, multiply the resulting volume by 10/9 to find the volume after removing one-tenth, and use the volume formula for a cylinder to solve for the diameter of the hole.

Step-by-step explanation:

To determine the diameter of the hole drilled through the center of the solid, we need to find the volume of the solid and then remove one-tenth of that volume. The solid is bounded by the equations z = 25e^-(x²+y²)/4, z = 0, and x² + y² = 16. To find the volume, we integrate the equation z = 25e^-(x²+y²)/4 over the region defined by x² + y² = 16 and z = 0.

We then multiply the resulting volume by 10/9 to find the volume after removing one-tenth. Finally, we use the volume formula for a cylinder, V = πr²h, to solve for the diameter of the hole.