A certain rectangular crate measures 8 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 on one of its six faces. What should the radius of the tank be if it is to be of the largest possible volume?
a.4
b.5
c.6
d.8
e.10
Hey Viper
The first thing one can do in this question is to narrow down the options. Since we have a cuboid of dimensions 12 feet*10 feet*8 feet, we cannot possibly have a cylinder with radius 6,8 or 10, because a radius of 6, 8 or 10 would mean a diameter of 12, 16 or 20, and one cannot possibly fit such a cylinder into the cuboid.
So, we are down to 2 options, a.4 and b.5
Considering the second option, if the radius is 5, the cylinder will have it base along the 10 feet and 12 feet side, with 8 feet as the length.
So, volume = (22/7) * r^2 * height
= (22/7) * 25 * 8
=(22/7) * 200
If we take the first option, the cylinder can have any of the sides as the base.
Taking base as 8 feet * 10 feet, height =12 feet
Volume = (22/7) * 4^2 * 12
=(22/7) * 192
Taking base as 8*12, height = 10 feet
Volume = (22/7) * 4^2 * 10
=(22/7) * 160
Similarly, taking base as 10 feet * 12 feet, height = 8 feet
and hence, Volume = (22/7) * 128
If we compare all the volumes obtained, the maximum value is 200 pi, and that is when the radius is 5 cm. hence, the answer is b.5
Hope this clarifies
YNWA
