Geometry

[talaangoshtari]

What is the volume of the largest cube that can fit entirely within a cylinder with a volume of 45Ï€ and a height of 5?

A. 27
B. 54√2
C. 125
D. 216√2
E. 432√2

[MartyMurray]

The cube with the largest volume will be the one with the longest edges. So we need to determine how the length of the edges of the cube is constrained by the dimensions of the cylinder.

The height of the cylinder is 5. So, the maximum height of the cube would be 5.

What about the diameter of the cylinder? The diameter of the cylinder is equal to the length of the diagonal of a face of the maximum cube that would fit within the circle of the cylinder.




The volume of a cylinder is πr²h. So the radius of the cylinder is √(45/5) = √9 = 3.

Thus the diameter of the cylinder and maximum diagonal of the cube is 6.

The length of the edges of the face of a cube with diagonal 6 are the lengths of the sides of a 45-45-90 triangle with hypotenuse 6. So the lengths of the edges are 3√2.

3√2 < 5

We can't fit a bigger cube by tilting the cube somehow, because if the cube were tilted, then the diagonals of the cube would not fit within the cylinder.

So, the maximum length of the edges of the cube is 3√2.

Volume of a cube is the cube of the length of the edges.

(3√2)³ = 54√2

The correct answer is B.