Question

An aquarium 7 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.) Show how to approximate the required work by a Riemann sum. (Let x be the height in meters below the top of the tank. Enter xi* as xi.)

Answer 1

8575 joules

Explanation:

We have that the volume is given as follows:

v = 7 * 1 * delta x

v = 7 * delta x

we know that the density: mass / volume

therefore the mass is:

mass = volume * density

replacing:

mass = 7 * delta x * 1000

mass = 7000 * delta x [kg]

now we know that force equals mass * acceleration:

F = 7000 * delta x * 9.8 m / s ^ 2

F = 68600 * delta x [N]

Now, we multiply by x meters, to calculate the work:

W = F * x

W = 68600 * delta x * x

W = 68600 * delta x [J]

We know that the portion of the chian x to x + delta x to be the following

limit when n tends to infinity of the sum of i = 1 up to n = infinity of 68600 * xi * delta x

therefore, the total work done to pump half of the water out of the aquarium is:

To the integral of 68600 * x * dx from 0 to 1/2, applying the integral we have:

68600 * x ^ 2/2 [x = 0 to x = 1/2]

68600 * ((1/2) ^ 2) / 2 - 68600 * (0 ^ 2) / 2 = 8575

That is to say that the work is 8575 joules