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Sagot :
To determine the volume of the sphere, let's start by gathering the given information and applying the formulas for the volumes of a cylinder and a sphere.
### Step 1: Understanding the volume formulas
We know the following standard formulas for volumes:
- The volume of a cylinder is [tex]\( V_{\text{cylinder}} = \pi r^2 h \)[/tex]
- The volume of a sphere is [tex]\( V_{\text{sphere}} = \frac{4}{3}\pi r^3 \)[/tex]
### Step 2: Given information
- The volume of the cylinder is [tex]\( 21 \, \text{m}^3 \)[/tex].
- The cylinder and the sphere have the same radius [tex]\( r \)[/tex] and the same height [tex]\( h \)[/tex].
### Step 3: Relating the height of the cylinder to the radius of the sphere
Given that the cylinder and the sphere have the same radius and height, the height of the cylinder [tex]\( h \)[/tex] translates directly to the diameter of the sphere, which means:
[tex]\[ h = 2r \][/tex]
### Step 4: Using the volume of the cylinder to find the radius
From the volume of the cylinder formula:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
[tex]\[ 21 = \pi r^2 h \][/tex]
Substitute [tex]\( h = 2r \)[/tex] into the equation:
[tex]\[ 21 = \pi r^2 (2r) \][/tex]
[tex]\[ 21 = 2\pi r^3 \][/tex]
### Step 5: Solving for the radius [tex]\( r \)[/tex]
Rearrange and solve for [tex]\( r \)[/tex]:
[tex]\[ r^3 = \frac{21}{2\pi} \][/tex]
[tex]\[ r = \left( \frac{21}{2\pi} \right)^{1/3} \][/tex]
### Step 6: Calculating the volume of the sphere
Using the radius [tex]\( r \)[/tex] we found, we can now find the volume of the sphere:
[tex]\[ V_{\text{sphere}} = \frac{4}{3}\pi r^3 \][/tex]
Since [tex]\( r^3 = \frac{21}{2\pi} \)[/tex]:
[tex]\[ V_{\text{sphere}} = \frac{4}{3}\pi \left( \frac{21}{2\pi} \right) \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4}{3}\pi \left( \frac{21}{2\pi} \right) \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \cdot \frac{21}{2} \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4 \cdot 21}{3 \cdot 2} \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{84}{6} \][/tex]
[tex]\[ V_{\text{sphere}} = 14 \, \text{m}^3 \][/tex]
Therefore, the volume of the sphere is [tex]\( 14 \, \text{m}^3 \)[/tex].
### Step 1: Understanding the volume formulas
We know the following standard formulas for volumes:
- The volume of a cylinder is [tex]\( V_{\text{cylinder}} = \pi r^2 h \)[/tex]
- The volume of a sphere is [tex]\( V_{\text{sphere}} = \frac{4}{3}\pi r^3 \)[/tex]
### Step 2: Given information
- The volume of the cylinder is [tex]\( 21 \, \text{m}^3 \)[/tex].
- The cylinder and the sphere have the same radius [tex]\( r \)[/tex] and the same height [tex]\( h \)[/tex].
### Step 3: Relating the height of the cylinder to the radius of the sphere
Given that the cylinder and the sphere have the same radius and height, the height of the cylinder [tex]\( h \)[/tex] translates directly to the diameter of the sphere, which means:
[tex]\[ h = 2r \][/tex]
### Step 4: Using the volume of the cylinder to find the radius
From the volume of the cylinder formula:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
[tex]\[ 21 = \pi r^2 h \][/tex]
Substitute [tex]\( h = 2r \)[/tex] into the equation:
[tex]\[ 21 = \pi r^2 (2r) \][/tex]
[tex]\[ 21 = 2\pi r^3 \][/tex]
### Step 5: Solving for the radius [tex]\( r \)[/tex]
Rearrange and solve for [tex]\( r \)[/tex]:
[tex]\[ r^3 = \frac{21}{2\pi} \][/tex]
[tex]\[ r = \left( \frac{21}{2\pi} \right)^{1/3} \][/tex]
### Step 6: Calculating the volume of the sphere
Using the radius [tex]\( r \)[/tex] we found, we can now find the volume of the sphere:
[tex]\[ V_{\text{sphere}} = \frac{4}{3}\pi r^3 \][/tex]
Since [tex]\( r^3 = \frac{21}{2\pi} \)[/tex]:
[tex]\[ V_{\text{sphere}} = \frac{4}{3}\pi \left( \frac{21}{2\pi} \right) \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4}{3}\pi \left( \frac{21}{2\pi} \right) \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \cdot \frac{21}{2} \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4 \cdot 21}{3 \cdot 2} \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{84}{6} \][/tex]
[tex]\[ V_{\text{sphere}} = 14 \, \text{m}^3 \][/tex]
Therefore, the volume of the sphere is [tex]\( 14 \, \text{m}^3 \)[/tex].
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