A sphere has center (0,0,0) and a radius of 5. Which of the following points lies on the sphere? Use the distance formula, but adjust it to be used fir three variables instead of two.
The distance formula in three dimensions is:
distance = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
For a point to lie on the sphere with center (0,0,0) and radius 5, it must satisfy the equation:
distance = sqrt(x^2 + y^2 + z^2) = 5
We can test each of the given points to see if they satisfy this equation:
A) (4,3,1)
distance = sqrt((4-0)^2 + (3-0)^2 + (1-0)^2) = sqrt(16 + 9 + 1) = sqrt(26) ≠ 5
B) (-5,0,0)
distance = sqrt((-5-0)^2 + (0-0)^2 + (0-0)^2) = sqrt(25) = 5
C) (0,-3,4)
distance = sqrt((0-0)^2 + (-3-0)^2 + (4-0)^2) = sqrt(9 + 16) = sqrt(25) = 5
D) (0,0,6)
distance = sqrt((0-0)^2 + (0-0)^2 + (6-0)^2) = sqrt(36) = 6 ≠ 5
Therefore, the points on the sphere are B) (-5,0,0) and C) (0,-3,4).