Find the equation of the surface.
The bottom hemisphere of a sphere centered at (4, -5, 0) with radius 10.
To find the equation of the surface of the bottom hemisphere of a sphere, you can use the equation for a sphere centered at (h, k, l) with radius r:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
In this case, the center of the sphere is (4, -5, 0) and the radius is 10.
So, the equation of the surface of the bottom hemisphere can be written as:
(x - 4)^2 + (y + 5)^2 + z^2 = 100
This equation represents all the points on the surface of the bottom hemisphere of the given sphere.
To find the equation of the surface, we need to determine the equation of the bottom hemisphere of the sphere.
The equation of a sphere can be written in the general form as:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
where (h, k, l) represents the center of the sphere and r is the radius.
In this case, the center of the sphere is (4, -5, 0) and the radius is 10.
So, the equation of the bottom hemisphere of the sphere can be written as:
(x - 4)^2 + (y + 5)^2 + (z - 0)^2 = 10^2
Simplifying this equation, we get:
(x - 4)^2 + (y + 5)^2 + z^2 = 100
Thus, the equation of the surface is (x - 4)^2 + (y + 5)^2 + z^2 = 100.
The equation of a spherical surface centred at (0,0,0) is
x^2+y^2+z^2=r^2
centred at (x0,y0,z0)
(x-x0)^2+(y-y0)^2+(z-z0)^2=r^2
substituting C(4,-5,0),
the equation of the spherical surface is
(x-4)^2+(y+5)^2+z^2=10^2=100
Assuming z-coordinate is positive=up, then
we have
(x-4)^2+(y+5)^2+z^2=100 (z<=0)
for the bottom hemisphere.