Find an equation of the largest sphere with center (5,2,7)(5,2,7) and is contained in the first octant. be sure that your formula is monic.

x² + y² + z² - 10x - 4y - 14z + 74 = 0

Step-by-step explanation:

The general equation of a sphere is (x-a)² + (y-b)² + (z-c)² = r²

Where x, y, and z are the coordinates of points on the surface of the sphere.

a, b, and c represents the center of the sphere

r is the radius of the sphere. Note that the radius is always the same for all points on the sphere,

In this equation, the radius r is the largest radius that stays in the octant.

In the given question, (5,2,7) is the center of the sphere.

Therefore, substitute this into the general equation to get:

(x-5)²+(y-2)²+(z-7)² = r² (i)

To find the radius r, we have to look at the distance from the center coordinate to each bounding planes xy-plane, xz-plane, and yz-plane.

The distance from the center to the xy-plane is the center of the z coordinate which is 7. The distance from the center to the xz-plane is the center of the y coordinate which is 2. The distance from the center to the yz-plane is the center of the x coordinate which is 5.

Therefore, to determine the radius contained in the first octant, we need to choose the smallest distance so as not to cross into a second octant. That will also be the largest possible radius for it not to cross into a different octant.

The smallest distance therefore is 2. So we substitute r = 2 into equation (i) above to get:

(x-5)²+(y-2)²+(z-7)² = 2²

(x-5)²+(y-2)²+(z-7)² = 4

Therefore (x-5)²+(y-2)²+(z-7)²-4 = 0  (ii)

A monic formula is a formula where the highest power of its single variable has a coefficient of 1.

Therefore, we expand equation (ii) in form of a monic formula to get

x² + y² + z² - 10x - 4y - 14z + 74 = 0

The highest power of x², y², and z² is 1

add all the numbers and check the ones place.

the answer is 486. as the ones place is more than 4, the estimation will be the next 10 ie, 490

35 is an outlier

step-by-step explanation: