Question

The volume of a cube with side length x is V(x) = x3. The volume of a sphere with radius x is shown in the graph. When x = 1, which volume is greater? 1) Volume of cube 2) Volume of sphere

Answer 1

When
`\(x = 1\),` the volume of the cube
`\(V_{\text{cube}}(1) = 1^3 = 1\)` cubic unit, and the volume of the sphere
`\(V_{\text{sphere}}(1) = (4)/(3)\pi(1^3) = (4)/(3)\pi\)` cubic units. Therefore, the volume of the sphere is greater.

Step-by-step explanation:

The volume \
`(V\)` of a cube with side length
`\(x\)` is given by the formula
`\(V_{\text{cube}}(x) = x^3\)` . Substituting
`\(x = 1\),` we find
`\(V_{\text{cube}}(1) = 1^3 = 1\)` cubic unit. On the other hand, the volume
`\(V\)` of a sphere with radius
`\(x\)` is given by the formula
`\(V_{\text{sphere}}(x) = (4)/(3)\pi x^3\).` Substituting
`\(x = 1\)` , we find
`\(V_{\text{sphere}}(1) = (4)/(3)\pi\)` cubic units.

Comparing the two volumes when
`\(x = 1\)` , we observe that
`\(V_{\text{sphere}}(1) = (4)/(3)\pi\)` is greater than
`\(V_{\text{cube}}(1) = 1\).` This is because the volume of a sphere increases at a faster rate than the volume of a cube as the radius or side length increases. The constant factor
`\((4)/(3)\pi\)` in the formula for the volume of a sphere leads to a more significant increase.

Understanding and comparing volumes of geometric shapes are essential in various fields, including physics, engineering, and architecture. The choice between a cube and a sphere may depend on factors such as efficiency in space usage or minimizing surface area for a given volume.