Answer :
To find the volume of the solid bounded below by the plane [tex]\(z = 1\)[/tex] and above by the sphere [tex]\(x^2 + y^2 + z^2 = 72\)[/tex], we can approach this problem by conceptualizing it as finding the volume of a spherical cap.
### Steps to Solve:
#### 1. Understanding the Problem
- The sphere has the equation [tex]\(x^2 + y^2 + z^2 = 72\)[/tex].
- The plane intersects the sphere at [tex]\(z = 1\)[/tex].
- We need to find the volume of the part of the sphere that lies above [tex]\(z = 1\)[/tex].
#### 2. Find the Radius and the Center of the Sphere
- The given equation of the sphere is [tex]\(x^2 + y^2 + z^2 = 72\)[/tex], indicating the sphere is centered at the origin [tex]\((0, 0, 0)\)[/tex] with a radius [tex]\(R\)[/tex] such that [tex]\(R^2 = 72\)[/tex]. Hence [tex]\(R = \sqrt{72} = 6\sqrt{2}\)[/tex].
#### 3. Find the Height of the Spherical Cap
- The height [tex]\(h\)[/tex] of the spherical cap is the distance from the plane [tex]\(z = 1\)[/tex] to the topmost point of the sphere, which is at [tex]\(z = R = 6\sqrt{2}\)[/tex].
Thus, the height [tex]\(h\)[/tex] is:
[tex]\[ h = 6\sqrt{2} - 1 \][/tex]
#### 4. Use the Formula for the Volume of a Spherical Cap
- The volume [tex]\(V\)[/tex] of a spherical cap with base area [tex]\(A\)[/tex] and height [tex]\(h\)[/tex] can be found using the formula:
[tex]\[ V = \frac{1}{3} \pi h^2 \left(3R - h\right) \][/tex]
#### 5. Substitute the Values
- Here, [tex]\(R = 6\sqrt{2}\)[/tex] and [tex]\(h = 6\sqrt{2} - 1\)[/tex]:
[tex]\[ \begin{aligned} V &= \frac{1}{3} \pi (h^2) (3R - h) \\ &= \frac{1}{3} \pi \left((6\sqrt{2} - 1)^2\right)\left(3 \cdot 6\sqrt{2} - (6\sqrt{2} - 1)\right) \\ &= \frac{1}{3} \pi \left(72 - 12\sqrt{2} + 1\right)\left(18\sqrt{2} - 6\sqrt{2} + 1\right) \\ &= \frac{1}{3} \pi \left(73 - 12\sqrt{2}\right)\left(12\sqrt{2} + 1\right) \end{aligned} \][/tex]
Simplifying within the equation:
[tex]\[ \begin{aligned} V &= \frac{1}{3} \pi \left((73 \cdot 12\sqrt{2} + 73) - (12\sqrt{2} \cdot 12\sqrt{2} + 12\sqrt{2})\right) \\ &= \frac{1}{3} \pi \left(876\sqrt{2} + 73 - 288 - 12\sqrt{2}\right) \\ &= \frac{1}{3} \pi \left(864\sqrt{2} + 73 - 288\right) \\ &= \frac{1}{3} \pi \left(864\sqrt{2} - 215\right) \end{aligned} \][/tex]
Since simplifying [tex]\(864\sqrt{2}\)[/tex]:
[tex]\[ \begin{aligned} Volume &= \frac{\pi}{3}\left(864\sqrt{2}-214\right) \end{aligned} \][/tex]
### Final Answer:
[tex]\( V = \boxed{\frac{864\sqrt{2}-215}{3} \pi} \)[/tex]
### Steps to Solve:
#### 1. Understanding the Problem
- The sphere has the equation [tex]\(x^2 + y^2 + z^2 = 72\)[/tex].
- The plane intersects the sphere at [tex]\(z = 1\)[/tex].
- We need to find the volume of the part of the sphere that lies above [tex]\(z = 1\)[/tex].
#### 2. Find the Radius and the Center of the Sphere
- The given equation of the sphere is [tex]\(x^2 + y^2 + z^2 = 72\)[/tex], indicating the sphere is centered at the origin [tex]\((0, 0, 0)\)[/tex] with a radius [tex]\(R\)[/tex] such that [tex]\(R^2 = 72\)[/tex]. Hence [tex]\(R = \sqrt{72} = 6\sqrt{2}\)[/tex].
#### 3. Find the Height of the Spherical Cap
- The height [tex]\(h\)[/tex] of the spherical cap is the distance from the plane [tex]\(z = 1\)[/tex] to the topmost point of the sphere, which is at [tex]\(z = R = 6\sqrt{2}\)[/tex].
Thus, the height [tex]\(h\)[/tex] is:
[tex]\[ h = 6\sqrt{2} - 1 \][/tex]
#### 4. Use the Formula for the Volume of a Spherical Cap
- The volume [tex]\(V\)[/tex] of a spherical cap with base area [tex]\(A\)[/tex] and height [tex]\(h\)[/tex] can be found using the formula:
[tex]\[ V = \frac{1}{3} \pi h^2 \left(3R - h\right) \][/tex]
#### 5. Substitute the Values
- Here, [tex]\(R = 6\sqrt{2}\)[/tex] and [tex]\(h = 6\sqrt{2} - 1\)[/tex]:
[tex]\[ \begin{aligned} V &= \frac{1}{3} \pi (h^2) (3R - h) \\ &= \frac{1}{3} \pi \left((6\sqrt{2} - 1)^2\right)\left(3 \cdot 6\sqrt{2} - (6\sqrt{2} - 1)\right) \\ &= \frac{1}{3} \pi \left(72 - 12\sqrt{2} + 1\right)\left(18\sqrt{2} - 6\sqrt{2} + 1\right) \\ &= \frac{1}{3} \pi \left(73 - 12\sqrt{2}\right)\left(12\sqrt{2} + 1\right) \end{aligned} \][/tex]
Simplifying within the equation:
[tex]\[ \begin{aligned} V &= \frac{1}{3} \pi \left((73 \cdot 12\sqrt{2} + 73) - (12\sqrt{2} \cdot 12\sqrt{2} + 12\sqrt{2})\right) \\ &= \frac{1}{3} \pi \left(876\sqrt{2} + 73 - 288 - 12\sqrt{2}\right) \\ &= \frac{1}{3} \pi \left(864\sqrt{2} + 73 - 288\right) \\ &= \frac{1}{3} \pi \left(864\sqrt{2} - 215\right) \end{aligned} \][/tex]
Since simplifying [tex]\(864\sqrt{2}\)[/tex]:
[tex]\[ \begin{aligned} Volume &= \frac{\pi}{3}\left(864\sqrt{2}-214\right) \end{aligned} \][/tex]
### Final Answer:
[tex]\( V = \boxed{\frac{864\sqrt{2}-215}{3} \pi} \)[/tex]
