A trough is 3 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of x^2 from -1 to 1 . The trough is full of water. Find the amount of work required to empty the trough by pumping the water over the top. Note: The weight of water is 62 pounds per cubic foot.
To find the amount of work required to empty the trough, we need to calculate the weight of the water in the trough and then convert it to work.
First, let's find the volume of the water in the trough. Since the vertical cross-section of the trough is shaped like the graph of x^2 from -1 to 1, we can find the cross-sectional area at any point x between -1 and 1 by calculating x^2.
The length of the trough is given as 3 feet, and the height is given as 1 foot. So, the volume of the water in the trough can be calculated by integrating the cross-sectional area from -1 to 1 and multiplying it by the length of the trough:
V = integral[from -1 to 1] (x^2) dx * length of the trough
V = integral[from -1 to 1] (x^2) dx * 3
The integral of x^2 with respect to x is (1/3)x^3. So, the volume of the water in the trough is:
V = [ (1/3)x^3 ] [from -1 to 1] * 3
V = (1/3)(1^3 - (-1)^3) * 3
V = (1/3)(1 - (-1)) * 3
V = (1/3)(2) * 3
V = 2
Since the trough is full of water, the volume of water in the trough is 2 cubic feet.
Next, we need to calculate the weight of the water. The weight of water is given as 62 pounds per cubic foot. So, the weight of the water in the trough is:
Weight = volume of water * weight per cubic foot
Weight = 2 * 62
Weight = 124 pounds
Now, to convert the weight of the water to work, we need to multiply it by the height that the water needs to be pumped. The height of the trough is given as 1 foot. So, the amount of work required to empty the trough is:
Work = weight of the water * height of the trough
Work = 124 * 1
Work = 124 foot-pounds
Therefore, the amount of work required to empty the trough by pumping the water over the top is 124 foot-pounds.