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

Answer:

$(x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = 16$

Step-by-step 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

$(x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = 4^{2}$

$(x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = 16$