Respuesta :
we know that
the equation of the sphere is equal to
[tex](x-h)^{2} +(y-k)^{2}+(z-l)^{2}=r^{2}[/tex]
where
(h,k,l) is the center of the sphere
r is the radius of the sphere
Step 1
Find the center of the sphere (h,k,l)
Let
A(8,7,6) B(9,9,9)
we know that
the midpoint of the diameter is the center point
Find the midpoint AB
in the x-coordinate
=(8+9)/2------> 8.5
in the y-coordinate
=(7+9)/2------> 8
in the z-coordinate
=(6+9)/2------> 7.5
the center is (8.5,8,7.5)
Step 2
Find the length of the diameter (distance AB)
A(8,7,6) B(9,9,9)
[tex]d=\sqrt{(x-x1)^{2}+(y-y1)^{2}+(z-z1)^{2}}[/tex]
[tex]dAB=\sqrt{(9-8)^{2}+(9-7)^{2}+(9-6)^{2}}[/tex]
[tex]dAB=\sqrt{(1)^{2}+(2)^{2}+(3)^{2}}[/tex]
[tex]dAB=\sqrt{14}\ units[/tex]
the diameter is [tex]\sqrt{14}\ units[/tex]
the radius is [tex]r=(\sqrt{14} )/2\ units[/tex]
Step 3
Find the equation of the sphere
[tex](x-h)^{2} +(y-k)^{2}+(z-l)^{2}=r^{2}[/tex]
[tex]r=(\sqrt{14} )/2\ units[/tex]
the center is (8.5,8,7.5)
Substitute
[tex](x-8.5)^{2} +(y-8)^{2}+(z-7.5)^{2}=(\sqrt{14} )/2)^{2}[/tex]
[tex](x-8.5)^{2} +(y-8)^{2}+(z-7.5)^{2}=(14/4)[/tex]
[tex](x-8.5)^{2} +(y-8)^{2}+(z-7.5)^{2}=(3.5)[/tex]
therefore
the answer is
The equation of the sphere is
[tex](x-8.5)^{2} +(y-8)^{2}+(z-7.5)^{2}=(3.5)[/tex]
we are given
diameters has endpoints (8,7,6) and (9,9,9)
Center:
we will find mid point of these two points
[tex](a,b,c)=(\frac{8+9}{2},\frac{7+9}{2},\frac{6+9}{2})[/tex]
[tex](a,b,c)=(8.5,8,7.5)[/tex]
Radius:
radius is half of distance between these two points
[tex]d=\sqrt{(9-8)^2+(9-7)^2+(9-6)^2}[/tex]
[tex]d=\sqrt{14}[/tex]
now, we can find radius
[tex]r=\frac{d}{2}[/tex]
[tex]r=\frac{\sqrt{14}}{2}[/tex]
Equation of sphere:
we can use formula
[tex](x-a)^2+(y-b)^2+(z-c)^2=r^2[/tex]
now, we can plug values
and we get
[tex](x-8.5)^2+(y-8)^2+(z-7.5)^2=(\frac{\sqrt{14}}{2})^2[/tex]
[tex](x-8.5)^2+(y-8)^2+(z-7.5)^2=\frac{14}{4}[/tex].............Answer
