What is the volume of the largest cylinder that can fit into a box of

Volume of a Cylinder is Pi\(r^2\)h

Lets consider 10 as the diameter of the cylinder. radius becomes 5.

height becomes 8.

Then Volume = Pi*25*8 = 200Pi.

Is my ans wrong?

You are taking length of box as 10, then width also needs to be at least 10 so that the cylinder fits the box, which is not the case in the given question.
You can take length as 8, so that radius=4. We should take width as 10 to fit the cylinder and height would be 6.

Then height=6
Volume= pi*4*4*6= 96*pi

If we take another case, in which we take length as 8 and 6, so that diameter=6, then height of cylinder=10

Then volume in this case:
V= pi * 3*3*10=90*pi

Hence 96*pi is the largest volume.

Dimesion of box = 6 by 8 by 10

for Cylinder to have maximum volume teh radius should be as large as possible

If the circular face of cylinder is placed on the face of box with dimension 8x10 then the diameter of cylinder may be 8 at the most

i.e. Radius = 8/2 = 4 and height = 6
Volume \(= πr^2*h = π*4^2*6 = 96π\)

Answer: option D

Afc0892

Yes, Your answer is wrong because of circular face lies on the rectangular face of dimension 6x10 then the maximum diameter may be 6 if height is taken as 8
There are three cases

1) Circular face of cylinder lies on face with dimension 6x8, then Diameter = 6 and Height = 10, Now Volume \(= π*r^2*h = π*3^2*10 = 90π\)

2) Circular face of cylinder lies on face with dimension 6x10, then Diameter = 6 and Height = 8, Now Volume \(= π*r^2*h = π*3^2*8 = 72π\)

3) Circular face of cylinder lies on face with dimension 8x10, then Diameter = 8 and Height = 6, Now Volume \(= π*r^2*h = π*4^2*6 = 96π\)

I hope this helps!!!

Understood. Thanks sir.

The largest cylinder that will fit into a box of dimensions 6 by 8 by 10 will have the diameter of its base equal to one of the dimensions of the box and the height equal to another dimension of the box. Furthermore, if the base of the cylinder rests on a face of the box that is a by b, then the diameter of the base can’t exceed the lesser of a and b. For example, if the base of the cylinder rests on a face of the box that is 6 by 8, then the diameter of the base can’t exceed 6. With this in mind, let’s explore all the possible options of the volume of the cylinder. Recall that the volume of a cylinder is V = πr^2h

1) The base rests on a face that is 6 by 8; thus, the diameter = 6 and hence the radius = 3 and height = 10.

V = π(3)^2(10) = 90π


2) The base rests on a face that is 6 by 10; thus, the diameter = 6 and hence the radius = 3 and height = 8.

V = π(3)^2(8) = 72π

3) The base rests on a face that is 8 by 10; thus, the diameter = 8 and hence the radius = 4 and height = 6.

V = π(4)^2(6) = 96π

We see that 96π is the largest possible volume for the cylinder.

Answer: D
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