Find the area of the surface. $\newline$ The part of the sphere $x^{2}+y^{2}+z^{2}=49$ that lies above the plane $z=2$ . $\newline$ Submit Answer

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Q. Find the area of the surface.

\newline

The part of the sphere

x^{2}+y^{2}+z^{2}=49

that lies above the plane

z=2

\newline

Submit Answer

Sphere Equation: The equation of the sphere is $x^2 + y^2 + z^2 = 49$ , which is a sphere with radius $r = 7$ centered at the origin $(0,0,0)$ .
Plane Intersection: The plane $z = 2$ cuts the sphere, creating a circular cap on the sphere. We need to find the area of this cap.
Radius Calculation: To find the radius of the cap, we can use the equation of the sphere with $z = 2$ . So we plug $z = 2$ into the sphere's equation: $x^2 + y^2 + (2)^2 = 49$ .
Circle Area: Simplify the equation: $x^2 + y^2 + 4 = 49$ .
Circle Radius: Subtract $4$ from both sides to find the radius of the circle at $z = 2$ : $x^2 + y^2 = 45$ .
Circle Area Calculation: The radius of the circle at $z = 2$ is the square root of $45$ , which is $\sqrt{45}$ or $3\sqrt{5}$ .
Spherical Cap Area Formula: The area of a circle is $\pi$ times the radius squared, so the area of the circle at $z = 2$ is $\pi(3\sqrt{5})^2$ .
Radius and Height: Simplify the area: $\pi(9 \times 5) = 45\pi$ .
Height Calculation: However, we need the surface area of the spherical cap, not just the circle. The formula for the surface area of a spherical cap is $2\pi rh$ , where $r$ is the radius of the base of the cap and $h$ is the height of the cap.
Surface Area Calculation: We already have the radius $r = 3\sqrt{5}$ . Now we need to find the height $h$ of the cap. The height $h$ is the distance from the plane $z = 2$ to the top of the sphere, which is the radius of the sphere minus the z-coordinate of the plane: $h = 7 - 2$ .
Correcting Mistake: Calculate the height $h$ : $h = 7 - 2 = 5$ .
Correcting Mistake: Calculate the height $h$ : $h = 7 - 2 = 5$ .Now we have the radius $r = 3\sqrt{5}$ and the height $h = 5$ . Plug these into the formula for the surface area of a spherical cap: Surface Area = $2\pi(3\sqrt{5})(5)$ .
Correcting Mistake: Calculate the height $h$ : $h = 7 - 2 = 5$ .Now we have the radius $r = 3\sqrt{5}$ and the height $h = 5$ . Plug these into the formula for the surface area of a spherical cap: Surface Area = $2\pi(3\sqrt{5})(5)$ .Simplify the surface area: Surface Area = $2\pi(15\sqrt{5}) = 30\sqrt{5}\pi$ .
Correcting Mistake: Calculate the height $h$ : $h = 7 - 2 = 5$ .Now we have the radius $r = 3\sqrt{5}$ and the height $h = 5$ . Plug these into the formula for the surface area of a spherical cap: Surface Area = $2\pi(3\sqrt{5})(5)$ .Simplify the surface area: Surface Area = $2\pi(15\sqrt{5}) = 30\sqrt{5}\pi$ .But wait, we made a mistake in the formula for the surface area of a spherical cap. The correct formula is $2\pi rh$ , but we need to use the radius of the sphere, not the radius of the circle at $z = 2$ . The radius of the sphere is $7$ , not $3\sqrt{5}$ . We need to correct this.