A cylindrical tank of r=5 and h=9 is 2/3 filled with a liquid with density [rho]lb/ft^3. Find the Work required to pump liquid over the rim.
The main thing I need help with is finding the y-value at which the tank is 2/3 full. Assume the bottom point of the tank is at y=0.
To find the y-value at which the tank is 2/3 full, we need to determine the height of the liquid inside the tank.
The total volume of a cylindrical tank is given by the formula V = πr^2h, where r is the radius and h is the height.
In this case, the radius of the tank is r = 5 ft and the total height is h = 9 ft.
Since the tank is 2/3 filled, we need to find the height of the liquid when the volume is 2/3 of the total volume.
Let's set up an equation to solve for the height of the liquid.
2/3 of the total volume = (2/3) * π * r^2 * h
Now, plug in the given values:
2/3 of the total volume = (2/3) * π * 5^2 * 9
Simplifying,
2/3 of the total volume = (2/3) * π * 25 * 9
2/3 of the total volume = (2/3) * 225 * π
2/3 of the total volume = 150 * π
Now, let's solve for the height:
π * r^2 * h = 150 * π
Simplifying,
r^2 * h = 150
Substituting the values:
5^2 * h = 150
25 * h = 150
h = 150 / 25
h = 6 ft
Therefore, the y-value at which the tank is 2/3 full is 6 ft.
To find the y-value at which the tank is 2/3 full, we need to determine the height of the liquid inside the tank.
The volume of the liquid in the cylinder can be calculated using the formula for the volume of a cylinder:
V = πr^2h
Given that the radius (r) is 5 and the height (h) is 9, we can substitute these values into the formula:
V = π(5^2)(9) = 225π ft^3
Since the tank is 2/3 filled, the volume of the liquid is 2/3 of the total volume of the tank:
V_liquid = (2/3) * (225π) = 150π ft^3
Now, we can set up an equation to find the height of the liquid (y) inside the tank. Since the tank is a cylinder, the volume of the liquid can also be calculated using the formula for the volume of a cylindrical segment:
V_liquid = πy^2 (3r - y)
Substituting the known values:
150π = πy^2 (3(5) - y)
Simplifying the equation:
150 = y^2 (15 - y)
We can now solve this equation for y. Rearrange the equation to get:
y^3 - 15y^2 + 150 = 0
You can solve this cubic equation using numerical methods or a graphing calculator to find the three possible values for y. However, since the height of the liquid should be positive and less than the total height of the cylinder (9 ft in this case), you can ignore any roots that do not satisfy these conditions.
Once you have determined the correct value of y, you can now proceed to calculate the work required to pump the liquid over the rim.