The smaller region cut from the solid
sphere p < 12 by the plane z = 6 is a
spherical cap. To find its volume, we can use the formula for the volume of a spherical cap, which is given by:
V = (1/3)nh^2(3R - h)
where V is the volume, his the height of the cap, and R is the radius of the sphere.
In this case, the equation of the
plane, z = 6, tells us that the height of
the cap is 6 units. The radius of the sphere can be found by considering that p < 12. In a spherical coordinate system, p represents the distance
from the origin to a point, so p = 12
gives us a sphere with a radius of 12 units.
Now we can substitute the values into the formula:
V = (1/3)1(6^2)(3(12) - 6)
Simplifying further:
V = (1/3)п(36)(36 - 6)
V = (1/3)п(36)(30)
V = (1/3)(п)(1080)
V = 1130.97 cubic units
Therefore, the volume of the smaller region cut from the solid sphere p<
12 by the plane z = 6 is approximately
1130.97 cubic units.
Hope this helped :)