What is the volume of the largest cube that can fit

Hi pushpitkc

We know \(a^2 + a^2 = 16.\)
So, \(a^3 = 16\sqrt{2}\)

But why can't the side length be 3. The Diameter of Cylinder is 4. So the cube can be easily fitted with its mid point passing through the centre of circle. Hence, lenght of edge can be 3.

Hi rahul16singh28

Imagine a square with length "3". The diagonal of the square will be \(3\sqrt{2} = 4.2\)(approximately)

The diameter of the circle is 4(as you have rightly pointed out) and the square will go outside the circle.

Therefore, the cube of length 3 cannot be placed inside the cylinder with a radius of "2".

Hope this helps you!

Hi rahul16singh28,

I will add to what pushpitkc has replied.

When we are imagining about cube with maximum are in a cylinder, we also need to consider how this cube would be inserted in cylinder.

Imagine. You have built cylinder first. Then you have to build a cube and place it inside cylinder.

Now proceed with your calculation.

If you try to insert a cube of side = 3. diagonal of cube becomes \(3\sqrt{2}\).

Note that when we try to insert a square in a circle, we always consider the diagonal of square. Isn't it.

Because diagonal of square will be equal to diameter of circle. And this ensures that square fits in the circle perfectly.

And E is definitely trap answer and most of us would select it, when we are short of time on the exam day.

Hope this will help you to understand how to approach the questions like this.

Let's first inscribe the largest possible square inside the circle



Since the radius of the cylinder is 2, we know that the DIAMETER = 4


Since we have a RIGHT TRIANGLE, we can apply the Pythagorean Theorem to get: x² + x² = 4²
Simplify: 2x² = 16
So, x² = 8, which means x = √8 = 2√2

ASIDE: On test day, you should know the following approximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2

So, we get: x = 2√2 ≈ 2(1.4) ≈ 2.8



At this point, we should recognize that, since the height of the cylinder is 3...

...then the LARGEST CUBE will have dimensions 2√2 by 2√2 by 2√2

Volume = (2√2)(2√2)(2√2)
= 8√8
= 8(2√2)
= 16√2

Answer: D

Cheers,
Brent

i get really confused in these. in some questions, you take a root 2 for cube and in others its a root 3. Can you pls explain


What is the maximum possible volume of a cube which can be placed inside a sphere with radius 10 units? (in unit^3).

pls explain in detail

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