To find the volumes of the sphere and the cube, we need a specific size for the sphere. However, since the dimensions of the sphere aren't provided, let's denote the radius of the sphere as \( r \). The relationship between the sphere and the cube is that the sphere fits perfectly inside the cube, meaning the diameter of the sphere equals the length of the side of the cube.
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Calculate the volume of the sphere: The formula for the volume \( V \) of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] Using \( \pi \approx 3.14 \), the formula becomes: \[ V \approx \frac{4}{3} \times 3.14 \times r^3 = \frac{12.56}{3} r^3 \approx 4.19 r^3 \]
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Calculate the volume of the cube: Since the diameter of the sphere is \( 2r \), the length of a side of the cube \( s \) is \( 2r \). The volume \( V \) of the cube is: \[ V = s^3 = (2r)^3 = 8r^3 \]
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Calculate the volume of the space between the sphere and the cube: This volume is the volume of the cube minus the volume of the sphere: \[ V_{\text{space}} = V_{\text{cube}} - V_{\text{sphere}} = 8r^3 - 4.19r^3 \] Simplifying this: \[ V_{\text{space}} = (8 - 4.19)r^3 = 3.81r^3 \]
Summary of Results:
- Volume of the Sphere: \( \approx 4.19 r^3 \)
- Volume of the Cube: \( 8 r^3 \)
- Volume of the Space: \( \approx 3.81 r^3 \)
To provide numerical values, we need the radius \( r \) of the sphere. If you provide a value for \( r \), we can compute the exact volumes and the space between them.
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