Calculate the area of the surface S.
-$S$ is the lower portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49$ cut by the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$.
A) $49 \left( \frac { \sqrt { 2 } } { 2 } + 1 \right)$
B) $14 \left( \frac { \sqrt { 2 } } { 2 } + 1 \right) \pi$
C) $98 \left( \frac { \sqrt { 2 } } { 2 } + 1 \right) \pi$
D) $49 \left( \frac { \sqrt { 2 } } { 2 } + 1 \right) \pi$

Question

Quizplus · Accepted Answer

The area of the surface $S$ can be calculated using the formula for the surface area of a sphere, $A = 4\pi r^2$, but since we are only considering the lower portion cut by the cone, we need to find the area of the spherical cap. The intersection of the sphere and the cone occurs where $z = \sqrt{x^2 + y^2} = \sqrt{49 - z^2}$, which simplifies to $2z^2 = 49$, giving $z = \frac{7}{\sqrt{2}}$. The height of the spherical cap is $h = r - z = 7 - \frac{7}{\sqrt{2}} = 7\left(1 - \frac{1}{\sqrt{2}}\right)$. The area of the spherical cap is $2\pi rh$, substituting $r = 7$ and $h = 7\left(1 - \frac{1}{\sqrt{2}}\right)$ gives $A = 2\pi \cdot 7 \cdot 7\left(1 - \frac{1}{\sqrt{2}}\right) = 98\pi\left(1 - \frac{1}{\sqrt{2}}\right)$, which simplifies to $98\pi\left(\frac{\sqrt{2}}{2} + 1\right)$.

Calculate the Area of the Surface S SSS Is the Lower Portion of the Sphere

Calculate the Area of the Surface S $S$ Is the Lower Portion of the Sphere