A tank is full of water. Find the work W required to pump the water out of the s

Question

A tank is full of water. Find the work W required to pump the water out of the spout. (Use 9.8 m/s^2 for g. Use 1000 kg/m^3 as the weight density of water. Assume that a = 4 m, b = 4 m, c = 18 m, and d = 1 m.) W = J Enhanced Feedback Please Try dividing the tank into thin horizontal slabs of height Delta x. Let x be the distance between each slab and the spout. If the top surface of a slab has area A(x), then the slab's volume is approximately A(x)Delta x. Use the volume to compute the force required to pump the water out, and multiply by x to compute the work required to pump the water out. The amount of work required to empty the tank can be written as a Riemann sum and converted into an integral.

Explanation / Answer

Water has a density of 1000 kg/cubic meter.
Work = distance * force
force = mass * acceleration
acceleration = 9.8
Work = distance * mass * acceleration
Work = 9.8*mass*distance
You must calculate the mass and at every level since it is constantly changing.
The distance is equal to 4, the length of the spout, plus x, the amount of the tank that has been emptied so far.
Work = 9.8*(4+x)*mass
The tank is in the shape of a triangle with width 4 and height 4. As the water level drops the width and height will stay in proportion since it will remain a similar triangle to the original. So the width is equal to the height, and the height is equal to 4 - x. The mass of the water at any level is equal to ((4-x)*18)*1000, because this is the area of the rectangle which is the very top layer of water. The 4-x is the width, the 18 is the length, and the 1000 is the density of water.
Take the integral from 0 to 4 (Starting from the top and ending at the bottom) of Work, then. You get Work = 9.8*(4+x)*mass = 9.8*(4+x)*((4-x)*18)*1000.
Take the constants out of the integral:
9.8*18*1000* integral from 0 to 4 of (4-x)(4+x)
18000*9.8*integral from 0 to 4 of (16 - x^2)
18000*9.8*(16x - x^3/3 |4, 0)
18000*9.8*(64 – 64/3)
6000*9.8*128
75264000 Newtons

7526400000 J

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.