Calculus 2

posted by .

An underground tank full of water has the following shape:
Hemisphere - 5 m radius. at the bottom
Cylinder - radius 5 m and height 10m in the middle
Circular cone radius 5 m and height 4 m at the top
The top of the tank is 2 m below the ground surface and is connected to the surface by a spout. find the work required to empty the tank by pumping all of the water out of the tank up to the surface.
density of water = 1000 kg/m^3
Gravity = 10 m/s^2

I am doing to where I have three parts to this question. I find the work of all of them then add the work done of all 3 together.. However, I cannot figure out how to find the work done for the hemisphere OR the circular cone. Please help me solve this out I have no idea where to start!

  • Calculus 2 -

    OK, Here is the hemisphere.

    we have a hemisphere with base 16 feet below ground and bottom 21 feet below ground.

    We need its volume and the distance of the cg below ground.

    The volume is easy, half a sphere
    (1/2) (4/3) pi r^3 = (2/3) pi 125 = 250 pi/3

    the centroid of a sphere is 3/8 r from the base as derived here:

    http://mathworld.wolfram.com/Hemisphere.html

    Therefore the center of mass of the hemisphere is
    21 +(3/8)5
    below earth
    therefore we must lift a weight of water of
    rho g (250 pi/3) a distance of (21+15/8) meters
    that is in Joules
    use rho = 10^3 kg/m^3 and g = 10 m/s^2

  • whoops base at 16 meters -

    rho g (250 pi/3) a distance of (16+15/8) meters

  • Calculus 2 -

    Now do the cone the same way
    base is at 6 meters
    volume = (1/3) pi r^2(4)
    cg is at [6 - (1/4)4] meters below ground

  • Get this now? -

    Can you do the rest now?

  • Calculus 2 -

    Why would you do the distance of hemisphere from 0 to 16, when we are doing just the hemipsher alone then adding it to the rest after.. wouldnt distance by 5-dy

  • Calculus 2 -

    You are lifting the water from the cg of the hemisphere all the way to the surface.
    that is 16 meters to the top of the hemisphere plus another 15/8 to the cg

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    A cylinder is inscribed in a right circular cone of height 8 and radius (at the base) equal to 2. What are the dimensions of such a cylinder which has maximum volume?
  2. Math - Calculus 2

    An underground tank full of water has the following shape: Hemisphere - 5 m radius. at the bottom Cylinder - radius 5 m and height 10m in the middle Circular cone radius 5 m and height 4 m at the top The top of the tank is 2 m below …
  3. Calculus 2 / Physics

    An underground tank full of water has the following shape 1) hemisphere of radius 5 m at the bottom 2) a cylinder of radius 5 m and height 10 m in the middle 3) a circular cone with base radius 5 and height 4 m at the top The top of …
  4. calculus 2 or physics

    An underground tank full of water has the following shape 1) hemisphere of radius 5 m at the bottom 2) a cylinder of radius 5 m and height 10 m in the middle 3) a circular cone with base radius 5 and height 4 m at the top The top of …
  5. Math

    An underground tank full of water has the shape of a circular cone with a base radius 5m and height 4m at the top. The top of the tankl is 2m below the ground-surface and is connected to the surface by a spout. Find the work required …
  6. calculus

    A tank consists of a right circular cylinder at height (h)surmounted by a hemisphere at a radius (r) with an outlet atthe top. The tank is full of water at density ρkg/m3. Find the work required to empty thetank.
  7. Math

    A hemisphere of radius 7 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the …
  8. math

    Suppose we pump water into an inverted right-circular cone tank at the rate of 6 cubic feet per minute. The tank has the height 9 ft and radius on the top is 8 ft. What is the rate at which the water level is rising when the water …
  9. Calculus

    A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 2 . A) What are the dimensions of such a cylinder which has maximum volume?
  10. math

    A right circular cone stands on a hemisphere. The base radius of the cone is equal to the radius of the hemisphere. If the base radius of the cone is 10 centimeters and its height is 15 centimeters,calculate the total volume of the …

More Similar Questions