Find the volume An ornament is carved from a rectangular block of wood which has a square base and a height of 24 cm. The ornament consists of two identical spheres and two identical cubes. The...

Find the volume

An ornament is carved from a rectangular block of wood which has a
square base and a height of 24 cm. The ornament consists of two identical
spheres and two identical cubes. The diameter of each sphere is equal to the length of the side of each cube. The ornament has the same width as the original block. Find the volume of the ornament.

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You should come up with the following substitution for the length of the side of square such that:

`l = x`

You need to evaluate the volume of rectangular block of wood such that:

`V = 24x^2`

The ornament is formed from two spheres and two cubes, hence, the volume of ornament may be evaluated such that:

`V_1 = 2V_s + 2V_c`

You need to evaluate the volume of the sphere, using the provided information, such that:

`V_s = (4/3)pi*x^3`

You need to evaluate the volume of cube such that:

`V_c = x^3`

`(4/3)pi*x^3 + x^3 = 24x^2 => x^3*(4/3 pi + 1) = 24 x^2`

`x = 24/(4/3 pi + 1)`

`V_1 = (8/3)pi*(24/(4/3 pi + 1))^3 + (24/(4/3 pi + 1))^3`

Hence, evaluating the volume of ornament yields `V_1 = (8/3)pi*(24/(4/3 pi + 1))^3 + 2*(24/(4/3 pi + 1))^3.`

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