A closed cylindrical tank is 9ft. long and 4ft. in diameter, when lying in horizontal position, the water is 3ft. deep. if the tank is in vertical position, what is the depth of the water in the tank?

Question

king · Answer

should we first find vol of water in the tank and then find depth?

king · Answer

jamesj?

jamesj · Answer

one sec.   drawing a diagram

jamesj · Answer

In both cases obviously the volume of the water is the same.  So we need to
- first find the volume of water when the tank is lying down
- then set it equal to the volume of water when the take in standing up

Here's a diagram of the tank lying down.  To calculate the volume, find the cross-sectional area, as indicated in the shade region of the diagram, and multiply it by the length 9 ft.  

To find that cross-sectional area, integrate the equation for a circle of radius r = 2 ft with the appropriate limits of integration.  I'll you figure that piece out yourself.  This is the heart of the problem.

For the tank standing up, the area is easy.  For a height of water h,
$$ Volume = (Area \ of \ circle \ radius \ 2 \ ft)(height) = (\pi 2^2) h = 4\pi h $$

anonymous · Answer

i got an answer of 7.24ft.

anonymous · Answer

i don't get this "Volume=(Area of circle radius 2 ft)(height)=(π22)h=4πh"

jamesj · Answer

The volume of a solid with the same cross-section, like a cylinder, is just the area of the cross-section times the height.  That's where that volume formula comes from.

anonymous · Answer

In Figure b: 
 Volume of water = Ab × h  
 90.99= π(2)^2× h; h = 7.24

jamesj · Answer

Right.

anonymous · Answer

90.99 is the volume of water in fig. a , when the cylindrical tank is lying horizontal.

anonymous · Answer

thanks for the help jamesj and king :)