Question

Find an equation of the largest sphere that is centered at (5,4,9) and has interior contained in the first octant.

Answer 1

`(x - 5)^(2) + (y - 4)^(2) + (z - 9)^(2) = 16`

Explanation:

The general equation of a sphere is as follows:

`(x - x_(c))^(2) + (y - y_(c))^(2) + (z - z_(c))^(2) = r^(2)`

In which the center is
`(x_(c), y_(c), z_(c))`, and r is the radius.

In this problem, we have that:

`x_(c) = 5, y_(c) = 4, z_(c) = 1`

So

`(x - 5)^(2) + (y - 4)^(2) + (z - 9)^(2) = r^(2)`

Interior contained in the first octant:

The first octant is bounded by:

The xy plane, in which z is 0. The distance from the center of the sphere to the xy plane is 9.

The xz plane, in which y is 0. The distance from the center of the sphere to the xz plane is 4.

The yz plane, in which x is 0. The distance from the center of the sphere to the yz plane is 5.

This means that if the radius is higher than four, the sphere will cross into a different octant.

So the radius for the largest sphere is 4.

The equation is

`(x - 5)^(2) + (y - 4)^(2) + (z - 9)^(2) = 4^(2)`

`(x - 5)^(2) + (y - 4)^(2) + (z - 9)^(2) = 16`