Question

The length of the diagonal of a wooden cube is 24 cm. The cube is cut into the cylinder of the

Answer 1

The volume of the cylinder is $384\pi \sqrt{3cm }^{2}$.

What is the diagonal of the cube?

• The diagonal that runs through the middle of a cube is it's main diagonal; the diagonal that runs along one of its faces is not. Any cube's major diagonal can be calculated by multiplying one side's length by the square root of three.
• Cos-1(2/3) is the angle formed between a cube's diagonal and the diagonal of one of its faces.

Body of the diagonal=24cm.

Assume the length of the side= a.

$a^{2} +(a^{2} +a^{2} )=24^{2}$

                   $a=\sqrt[8]{3} cm$

The area of the cylinder base=$\pi R^{2}$

                                                =$\pi (\frac{1}{2} \sqrt[8]{3} )^{2}$

                                                =$48cm^{2}$

The high of the cylinder is$\sqrt[8]{3}cm$

The volume of the cylinder:

=s*h

=$48\pi *\sqrt[8]{3}$

=$384\pi \sqrt{3} cm^{3}$

The volume of the cylinder is $384\pi \sqrt{3cm }^{2}$.

To learn more about the diagonal of a cube, refer to:

brainly.com/question/14193105

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Answer 2

The biggest possible volume of the cylinder will be $2089.497\hspace{1mm}\text{cm}^3$.

What will be the diameter and height of the cylinder obtained from a cube and what are the formulas for the diagonal of a cube and the volume of a cylinder?

• If a cylinder with the biggest possible volume is cut inside the cube, the height of the cylinder and the diameter of the cylinder will be equal to the side length of the cube.
• For example, consider the following figure in which the cylinder is cut inside of the cube and since the side length of the cube is $x$, the diameter and the height of the cylinder are also $x$
• If the side length of a cube is $x$ unit, then its diagonal will be $x\sqrt{3}$ unit.
• The formula for the volume of a cylinder is $V=\pi r^2h$, where $r$ is the radius and $h$ is the height of the cylinder. If $d$ is the diameter, then $r=\frac{d}{2}$.

Now, given that the diagonal of the cube is $24$ cm. So, if the side length of the cube is $x$ cm, then we must have

$x\sqrt{3}=24\\\Longrightarrow x=\frac{24}{\sqrt{3}}\\\Longrightarrow x=8\sqrt{3}$

Thus, the side length of the cube is $8\sqrt{3}$ cm.

So, the height of the cylinder with maximum volume will be $h=8\sqrt{3}$ cm and the diameter will be $d=8\sqrt{3}$cm i.e. the radius will be $r=\frac{d}{2}=\frac{8\sqrt{3}}{2}=4\sqrt{3}$ cm.

So, using the above formula for the volume of a cylinder, we get

$V=\pi r^{2}h=\pi\times (4\sqrt{3})^2\times 8\sqrt{3}=2089.497\hspace{1mm}\text{cm}^3$.

Therefore, the biggest possible volume of the cylinder will be $2089.497\hspace{1mm}\text{cm}^3$.

To know more about volume, refer: brainly.com/question/1972490

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