(a) express the work as an integral (b) evaluate the integral

Question

An aquarium 3m long, 1m wide, and 1m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use 9.8m/s^2 for g and the fact that the density of the water is 1000 kg/m^3)

(a) express the work as an integral

(b) evaluate the integral

Answer 1

Solution:

Set up the vertical y-axis pointing upward so that the origin is at the center of aquarium. Let g = 9.8 m/s^2 and ρ = 1000 kg/m^3. For any y between 0 and 1/2, pick a typical slice Δy. Then the work done for this typical slice is

ΔW ≈ Weight * Height

= ρg*(3*1*Δy) * (1/2 - y)

So dW = 3ρg(1/2 - y) dy

Therefore, the work required is

W = int_0^(1/2) 3ρg(1/2 - y) dy

= 3ρg*(1/2 y -1/2 y^2) |_0^(1/2)

= 3/8 ρg

= 3/8 * 1000* 9.8

= 3675

Answer: The work needed to pump half of the water out of the aquarium is 3675 joules.