A trough is 10 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of y=x10 from x=−1 to x=1 . The trough is full of water. Find the amount of work in foot-pounds 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 by pumping the water over the top, we need to calculate the volume of water in the trough and then multiply it by the weight of the water. We can follow these steps:
Step 1: Determine the shape of the cross-section
The cross-section of the trough is described by the graph of y = x^10 from x = -1 to x = 1. This means that the width of the trough varies depending on the position along the length.
Step 2: Find the width of the trough at each position
To find the width of the trough at any position x, we can evaluate the function y = x^10. For the given cross-section, the width will vary from y = 0 to y = 1, following the shape of the graph. We can calculate the width by subtracting the y-values at each end.
Width at x: w = 1 - 0 = 1
Step 3: Find the height of the trough
The height of the trough is given as 1 foot.
Height: h = 1
Step 4: Find the length of the trough
The length of the trough is given as 10 feet.
Length: l = 10
Step 5: Calculate the volume of water in the trough
The volume of water in the trough can be calculated using the formula:
Volume = Length × Width × Height
In this case, we'll integrate the volume over the length of the trough. The volume is the integral of the area from x = -1 to x = 1:
Volume = ∫[x=-1 to 1] (Width(x) × Height) dx
Using the values from Step 2 and Step 3, we have:
Volume = ∫[x=-1 to 1] (1 × 1) dx
Simplifying the integral:
Volume = ∫[x=-1 to 1] dx
Integrating with respect to x:
Volume = [x] from -1 to 1
Volume = (1) - (-1)
Volume = 2
So, the volume of water in the trough is 2 cubic feet.
Step 6: Find the weight of water
The weight of water is given as 62 pounds per cubic foot.
Weight of water = Volume × Weight per cubic foot
Weight of water = 2 × 62
Weight of water = 124 pounds
Step 7: Calculate the work required to pump the water over the top
The work required to pump the water over the top can be found by multiplying the weight of water by the height the water needs to be lifted.
Work = Weight of water × Height
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.
To find the amount of work required to empty the trough, we need to calculate the volume of water in the trough and then determine the work done to lift this volume of water over the top.
Step 1: Determine the shape of the cross-section
The vertical cross-section of the trough is shaped like the graph of y = x^10 from x = -1 to x = 1. This means that the cross-section is symmetric about the y-axis and has a maximum height of 1 foot at x = 0.
Step 2: Determine the equation of the cross-sectional area
Since the cross-section is symmetric about the y-axis, we can find the area of the cross-section by integrating twice the positive portion. The cross-sectional area A(x) at a given x can be calculated by integrating the function y = x^10 from 0 to x and then doubling the result.
A(x) = 2 * ∫(0 to x) x^10 dx
Step 3: Calculate the volume of water
The volume of water in the trough can be found by integrating the cross-sectional area with respect to x over the length of the trough.
V = ∫(0 to 10) A(x) dx
Step 4: Convert volume to weight
The weight of water is 62 pounds per cubic foot. We can convert the volume of water (in cubic feet) to weight (in pounds) by multiplying by 62.
Weight = V * 62
Step 5: Calculate the work done
The work done to lift the water over the top of the trough is given by the equation: work = force * distance
Since we are lifting the water vertically by 1 foot, the distance is 1 foot. The force required to lift the weight of the water is equal to the weight itself.
Work = Weight * Distance
= Weight * 1
= Weight
So the amount of work required to empty the trough is equal to the weight of the water.
Therefore, the amount of work in foot-pounds required to empty the trough by pumping the water over the top is equal to the weight of the water, which is V * 62 pounds.