A tank has cross-sections in the form of isosceles triangels, vertex down. The top of the tank is 6ft wide, its height is 4ft, and its length is 10ft. find the work in pumping all the water to the top of the tank. (weight of water is 62.4 pounds per cubic feet)
here is my solution:
F = 62.4(volume) = 62.4(3)(91^0.5)dy
for work integrate 4(62.4(3)(91^0.5)dy
with the boudaries of integration from y=0 to y=4
To find the work required to pump all the water to the top of the tank, you need to calculate the integral of the force function over the appropriate interval.
The force function can be found by multiplying the weight of water per cubic foot (62.4 pounds) by the volume of each small slice of water.
First, let's determine the volume of each small slice. Since the cross-sections of the tank are isosceles triangles, we can use the formula for the area of a triangle, A = (1/2)base * height. In this case, the "base" of each triangle is changing as we move up the tank, but the height remains the same.
Given that the top of the tank is 6 feet wide and the height is 4 feet, we can write the base of each triangle as b = 6 - (6/4)y, where y represents the height from the base of the tank.
The volume of each small slice can be found by multiplying the base by the height and then by the small change in y, which we can denote as dy. Thus, the volume of each slice is dV = (1/2)(6 - (6/4)y)(4)dy = 3(6 - (3/2)y)dy.
Now, let's find the force function. We can calculate the weight of water for each slice by multiplying the volume, dV, by the weight per cubic foot, 62.4 pounds. Therefore, the force function is given by F = 62.4 * 3(6 - (3/2)y)dy.
To find the total work, we integrate the force function over the appropriate interval. Since we want to pump all the water to the top of the tank, we need to integrate with respect to y from the bottom of the tank (y = 0) to the top of the tank (y = 4).
Therefore, the work required to pump all the water to the top of the tank is given by the integral of 4(62.4 * 3(6 - (3/2)y))dy, with the boundaries of integration from y = 0 to y = 4.