Question

Find the work in ft-lbs required to empty a right cylindrical tank with a radius of 8 ft, a height of 6 ft, and a water level of 4 ft by pumping the water to the top of the tank

Answer 1

@sleepyhead314 @jigglypuff314

Answer 2

@Jesstho.-.

Answer 3

huh i don't actually know this one

Answer 4

i have some notes

Answer 5

do you want to see?

Answer 6

First we must find area of each horizontal slice at depth x (from top of tank), where x is between 0 and 6.

Draw a diagram:
http://oi60.tinypic.com/11m7w5w.jpg

Each horizontal slice is a circle with radius r.
Using Pythagorean theorem, we get:
x² + r² = 6²
r² = 36 − x²

Area of slice = πr² = π(36−x²) ft²
Thickness of slice = Δx ft
V = π(36−x²) Δx ft³

Water weighs 62.5 lbs/ft³. Now we calculate force:
F = 62.5 lbs/ft³ * π(36−x²) Δx ft³
F = 62.5 π (36−x²) Δx lbs

Finally, we find work to pump each slice out of the tank. Each slice has to be lifted a distance of x ft to top of tank, plus an additional 2 ft above that. So total distance is 2+x
W = F * d
W = (62.5 π (36−x²) Δx) lbs * (2+x) ft
W = 62.5 π (36−x²) (2+x) Δx ft-lbs

To find total work to empty tank, integrate from x = 0 to x = 6
W = ∫ [0 to 6] 62.5 π (36−x²) (2+x) dx ft-lbs
W = 62.5 π ∫ [0 to 6] (72+36x−2x²−x³) dx ft-lbs
W = 62.5 π (72x + 18x² − 2/3 x³ − 1/4 x⁴) | [0 to 6] ft-lbs
W = 62.5 π (612−0) ft-lbs
W = 38250π ft-lbs

Answer 7

this is for a hemisphere

Answer 8

@satellite73 @sourwing

Answer 9

@wio

Answer 10

64000pi