Calculate the area of the surface S.
-$S$ is the portion of the plane $7 x + 8 y + 7 z = 4$ that lies within the cylinder $x ^ { 2 } + y ^ { 2 } = 1$.
A) $\frac { 1 } { 7 } \pi$
B) $\frac { 18 } { 7 } \sqrt { 2 } \pi$
C) $\frac { 9 } { 7 } \sqrt { 2 } \pi$
D) $\frac { 9 } { 7 } \sqrt { 2 }$

Question

Quizplus · Accepted Answer

The area of the surface can be found by projecting the cylinder's circular base onto the plane and calculating the area of the resulting ellipse. The normal vector of the plane is $n = (7, 8, 7)$, and its magnitude is $\|n\| = \sqrt{7^2 + 8^2 + 7^2} = \sqrt{49 + 64 + 49} = \sqrt{162}$. The area of the circle in the cylinder's base is $\pi r^2 = \pi$, where $r = 1$. The area of the ellipse on the plane is the area of the circle multiplied by the cosine of the angle between the plane's normal and the z-axis. The cosine of this angle is given by the dot product of the normal vector and the z-axis unit vector $(0, 0, 1)$ divided by the magnitude of the normal vector, which simplifies to $7/\sqrt{162}$. However, the correct approach involves recognizing that the area scales by the factor of $1/\cos(\theta)$, where $\theta$ is the angle between the plane's normal and the z-axis, leading to scaling by $\sqrt{162}/7$. Thus, the area of the ellipse is $\pi \times (\sqrt{162}/7) = \pi \times \sqrt{162}/7$, which simplifies to $\frac{9}{7}\sqrt{2}\pi$, matching option C.

Calculate the Area of the Surface S SSS Is the Portion of the Plane

Calculate the Area of the Surface S $S$ Is the Portion of the Plane