Give a geometric description of the set of points whose coordinates satisfy the given conditions.
-$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49 , z = 1$$
A) All points within the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49$ and above the plane $z = 1$
B) All points on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49$ and above the plane $z = 1$
C) The sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$
D) The circle $x ^ { 2 } + y ^ { 2 } = 48$ in the plane $z = 1$

Question

Quizplus · Accepted Answer

The equation $x^2 + y^2 + z^2 = 49$ describes a sphere with radius 7 centered at the origin. Setting $z = 1$ slices this sphere with a plane parallel to the xy-plane, one unit above the origin, resulting in a circle. Substituting $z = 1$ into the sphere's equation gives $x^2 + y^2 + 1^2 = 49$, simplifying to $x^2 + y^2 = 48$, which is a circle with radius $\sqrt{48}$ in the plane $z = 1$.

Give a Geometric Description of the Set of Points Whose x2+y2+z2=49,z=1x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49 , z = 1x2+y2+z2=49,z=1

Give a Geometric Description of the Set of Points Whose $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49 , z = 1$