Answer :
Sure, let's walk through the problem step by step. You are given four spheres of radius 6 cm that fit perfectly inside a cylinder. You need to find the volume inside the cylinder that is empty.
### Step 1: Calculate the volume of one sphere
The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
For a sphere with radius [tex]\( r = 6 \)[/tex] cm:
[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]
[tex]\[ V = \frac{4}{3} \pi (216) \][/tex]
[tex]\[ V = \frac{864}{3} \pi \][/tex]
[tex]\[ V = 288 \pi \][/tex]
So, the volume of one sphere is [tex]\( 288 \pi \, \text{cm}^3 \)[/tex].
### Step 2: Calculate the total volume of four spheres
Since you have four spheres, the total volume will be:
[tex]\[ V_{\text{total spheres}} = 4 \times 288 \pi \][/tex]
[tex]\[ V_{\text{total spheres}} = 1152 \pi \, \text{cm}^3 \][/tex]
### Step 3: Determine the dimensions of the cylinder
- Height of the cylinder: The four spheres are stacked on top of each other, so the height of the cylinder will be four times the diameter of one sphere.
The diameter of one sphere is [tex]\( 2 \times r = 2 \times 6 = 12 \)[/tex] cm.
Therefore, the height of the cylinder is:
[tex]\[ \text{Height} = 4 \times 12 = 48 \, \text{cm} \][/tex]
- Radius of the cylinder: The radius of the cylinder is the same as the radius of one sphere, which is 6 cm.
### Step 4: Calculate the volume of the cylinder
The formula for the volume of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
For our cylinder with radius [tex]\( r = 6 \)[/tex] cm and height [tex]\( h = 48 \)[/tex] cm:
[tex]\[ V = \pi (6)^2 (48) \][/tex]
[tex]\[ V = \pi (36) (48) \][/tex]
[tex]\[ V = 1728 \pi \, \text{cm}^3 \][/tex]
### Step 5: Calculate the empty volume inside the cylinder
To find the empty volume, we subtract the total volume of the spheres from the volume of the cylinder:
[tex]\[ V_{\text{empty}} = V_{\text{cylinder}} - V_{\text{total spheres}} \][/tex]
[tex]\[ V_{\text{empty}} = 1728 \pi - 1152 \pi \][/tex]
[tex]\[ V_{\text{empty}} = 576 \pi \, \text{cm}^3 \][/tex]
### Final Answer
The volume inside the cylinder that is empty is:
[tex]\[ \boxed{576 \pi \, \text{cm}^3} \][/tex]
### Step 1: Calculate the volume of one sphere
The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
For a sphere with radius [tex]\( r = 6 \)[/tex] cm:
[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]
[tex]\[ V = \frac{4}{3} \pi (216) \][/tex]
[tex]\[ V = \frac{864}{3} \pi \][/tex]
[tex]\[ V = 288 \pi \][/tex]
So, the volume of one sphere is [tex]\( 288 \pi \, \text{cm}^3 \)[/tex].
### Step 2: Calculate the total volume of four spheres
Since you have four spheres, the total volume will be:
[tex]\[ V_{\text{total spheres}} = 4 \times 288 \pi \][/tex]
[tex]\[ V_{\text{total spheres}} = 1152 \pi \, \text{cm}^3 \][/tex]
### Step 3: Determine the dimensions of the cylinder
- Height of the cylinder: The four spheres are stacked on top of each other, so the height of the cylinder will be four times the diameter of one sphere.
The diameter of one sphere is [tex]\( 2 \times r = 2 \times 6 = 12 \)[/tex] cm.
Therefore, the height of the cylinder is:
[tex]\[ \text{Height} = 4 \times 12 = 48 \, \text{cm} \][/tex]
- Radius of the cylinder: The radius of the cylinder is the same as the radius of one sphere, which is 6 cm.
### Step 4: Calculate the volume of the cylinder
The formula for the volume of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
For our cylinder with radius [tex]\( r = 6 \)[/tex] cm and height [tex]\( h = 48 \)[/tex] cm:
[tex]\[ V = \pi (6)^2 (48) \][/tex]
[tex]\[ V = \pi (36) (48) \][/tex]
[tex]\[ V = 1728 \pi \, \text{cm}^3 \][/tex]
### Step 5: Calculate the empty volume inside the cylinder
To find the empty volume, we subtract the total volume of the spheres from the volume of the cylinder:
[tex]\[ V_{\text{empty}} = V_{\text{cylinder}} - V_{\text{total spheres}} \][/tex]
[tex]\[ V_{\text{empty}} = 1728 \pi - 1152 \pi \][/tex]
[tex]\[ V_{\text{empty}} = 576 \pi \, \text{cm}^3 \][/tex]
### Final Answer
The volume inside the cylinder that is empty is:
[tex]\[ \boxed{576 \pi \, \text{cm}^3} \][/tex]
