Find an equation of a sphere if one of its diameters has endpoints (4, 1, 6) and (8, 3, 8).

The equation of a sphere with endpoints at (4, 1, 6) and (8, 3, 8) is .

Step-by-step explanation:

Given the extremes of the diameter of the sphere, its center is the midpoint, whose location is presented below:

Any sphere with a radius and centered at is represented by the following equation:

Let be and , the radius of the sphere is now calculated:

The equation of a sphere with endpoints at (4, 1, 6) and (8, 3, 8) is .

if x decreases by 4 then y increases by 10

step-by-step explanation:

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt