What is the Correct Calculation for Pressure on a Dam?

[1MileCrash]

Homework Statement

Set up and calculate a definite integral giving the total pressure on the dam shown in the figure, which is about the size of the Aswan Dam in Egypt.

The figure is a trapezoid, top 3600m, bottom 3000m, height of 100m.

Homework Equations

The Attempt at a Solution

First, I will make horizontal slices since the pressure of the water in a horizontal slice is constant when the depth of the slice approaches 0.

Next, I want the area of a slice. To do this, I essentially broke it into three shapes. The two outer triangles, and the inner square. These make up the width, while delta y is the height.

The width of the inner square is a constant 3000. The triangles mimic the line x = 6y, and there are two of them, so I define the area of a slice to be:

(3000+12y)Δy

The pressure on a given slice varies with depth. The force on a given slice is the pressure multiplied by area.

1000gy(3000+12y)Δy

Now I add up all slices with integration, over a depth of 100 meters:

$\int^{100}_{0} 9800y(3000+12y)dy$

For a final answer of 1.862x10^11 Newtons.

This is incorrect. Where did I go wrong?

Thank you.


EDIT: Is it because the direction of increasing pressure and of "larger width" of the slices don't match? Must be it, how to correct? -12y?


Making that change, I get 1.07x10^11 N. I only have one attempt left, look good?

 

[mighty2000]

Your calculation looks good, but there are a few things you can improve upon. First, make sure you are using the correct units for all of your variables. In this case, the height of the dam is given in meters, so the units for Δy should also be in meters. Secondly, when setting up the integral, make sure you are using the correct limits of integration. In this case, since the depth of the dam is 100m, the upper limit should be 100, not 3600. Finally, when calculating the area of the slice, you should use the formula for the area of a trapezoid, which is 0.5*(base 1 + base 2)*height. This will give you a more accurate result. Keep up the good work!