# Math Help - Another Work Problem

1. ## Another Work Problem

There is a cylindrical gas tank (42lbs/ft3)4ft tall and with a diameter of 3ft. It is being carried on the back of a truck, horizontally. How much work would it take to empty the entire tank to another tank 5ft above the top of the original tank?

I originally tried solving with the tank lying horizontally but it didn't look like it would be easy trying to find a function for the volume of prices sliced horizontally. I reasoned that because the volume was the same, it wouldn't be different to just think of the tank ad being vertical. I did the problem that way and was incorrect which leads me to believe that my assumption about its orientation not mattering to be wrong.

I would really appreciate some assistance with this.
The correct answer is 2457pi and I got 2646pi.

2. ## Re: Another Work Problem

Well, your "reasoning" is wrong. The volume is the same but the distance the gas has to be "lifted" is different. With the tank on its side, every horizontal "slice" of gas is a rectangle with length equal to the length of the tank, 4 feet, and width equal to the length of a chord of the circle. If we set up a coordinate system with center at the center of circle, its equation is $x^2+ y^2= 9$. The width of the circle at a given y is $2x= 2\sqrt{9- y^2}$. Taking "dy" as the thicknes of a "slice", its volume is $4(2\sqrt{9-y^2})(dy)= 8\sqrt{9- y^2}dy$. Multiply that by the density of the gas and the height that "slice" has to lifted, 8- y (the top of the tank is at y= 3 and it has to be lifted another 5 feet), and integrate from y= -3 to y= 3.

3. ## Re: Another Work Problem

The diameter is 3ft, not the radius, but I understand your point.