kartikshah wrote:A certain rectangular crate measures 08 feet by 10 feet by 12 feet. A cylindrical gas tank is to be made for shipment in the crate and will stand upright when the crate is placed in one of its six faces. What should the radius of the tank be if it is to be of the largest possible volume?

1. 4
2. 5
3. 6
4. 8
5. 10

Thanks!
Source: Princeton Review Practice Test 10

As you are aware that Volume of a Cylinder V=Pie.r^2.h; where r= radius= diameter/2, & h= height.

So the objective is to maximise V or maximise(Pie.r^2.h);

Here Pie is constant.There are 2 variables r, & h need to be calculated.

We can also observe that Volume is proportional to height, but it is proportional to square of radius.

So the aim to assign maximum possible values out of 8,10, & 12 to radius.

Lets take 3 scenarios...

Case 1- Base is 8*10--- This option will render radius as 4 (8/2);

Case 2- Base is 12*10--- This option will render radius as 5 (10/2);

Case 3- Base is 8*12--- This option will render radius as 4 (8/2);

It is obvious that largest volume is possible if we make base as 12*10, & height as 8 ft.

So radius would be 5 ft., & Max. Volume = pie.5^2.8 = 200 cubic feet.