The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ lying within $x ^ { 2 } + y ^ { 2 } = 9$ is

A) $\pi ( 144 - 72 \sqrt { 3 } )$
B) $\pi ( 144 - 72 \sqrt { 5 } )$
C) $\pi ( 144 - 71 \sqrt { 3 } )$
D) $\pi ( 144 - 71 \sqrt { 5 } )$
E) $\pi ( 144 - 70 \sqrt { 3 } )$

Question

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ lying within $x ^ { 2 } + y ^ { 2 } = 9$ is

A) $\pi ( 144 - 72 \sqrt { 3 } )$
B) $\pi ( 144 - 72 \sqrt { 5 } )$
C) $\pi ( 144 - 71 \sqrt { 3 } )$
D) $\pi ( 144 - 71 \sqrt { 5 } )$
E) $\pi ( 144 - 70 \sqrt { 3 } )$

Quizplus · Accepted Answer

The surface described is a sphere with radius 6, and the condition $x^2 + y^2 = 9$ describes a cylinder that intersects the sphere. The surface area of the sphere within the cylinder can be found by calculating the area of the spherical cap cut off by the cylinder. The height of the cap, $h$, can be found using the Pythagorean theorem, where $r^2 + h^2 = 36$ and $r = 3$ (the radius of the cylinder), giving $h = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3}$. The surface area of the spherical cap is $2\pi R h = 2\pi(6)(3\sqrt{3}) = 36\pi\sqrt{3}$, where $R = 6$ is the radius of the sphere. The total surface area of the sphere is $4\pi R^2 = 4\pi(6)^2 = 144\pi$. The area of the sphere outside the cylinder is the total area minus the area of the cap, which is $144\pi - 36\pi\sqrt{3} = \pi(144 - 36\sqrt{3})$. However, since the cap is on both the top and bottom, we subtract twice the area of the cap, leading to the correct answer of $\pi(144 - 72\sqrt{3})$.

The Surface Area of the Surface x2+y2+z2=36x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36x2+y2+z2=36

The Surface Area of the Surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$