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Problem 16 Medium Difficulty

A dam is inclined at an angle of $ 30^\circ $ from the vertical and has the shape of an isosceles trapezoid 100 ft wide at the top and 50 ft wide at the bottom and with a slant height of 70 ft. Find the hydrostatic force on the dam when it is full of water.

Answer

$8.832 \times 10^{6} \mathrm{lbs}$

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Video Transcript

in this problem were given not a done. Uh, is it Trump? Is it'll shape and its banks 30 degrees with the working girl. So where has to find a total force acting on a side off this, then we know that total forces it with pressure times area and is able to rotate g times that pleasure that the reference multiplied by area. We don't have four water, rock G or white dance This 62 £15 for feet. Cute. So we notice. We know that, uh, we need to find an expression for that. And for the area or it. We know that water is acting on this surface. It means that we would like to find a area off this. But as you can see as the step measured here as that, that changes the with every Why changes. It means that the area will change as well. In order to find aerialist. First issue, not we haven't origin located right here. Sentry duty for Be right here. That is our XX, sis. This is R y axis. So this would be X and it this would be our Y X is so this is 00 co ordinates off this point right here with them. We don't forget that this ship is semantic inspectorswork Alexis. So they totaling is 100. This would be 50 0 and the coordinates off this word point would then be 25 by 70. Why do we need the coordinates off those two? Because we would like to ride equation off this following line so that we could find an expression for wit is a function of what? 25 70 Be not only question off the line, you will for why minus Why not is a slope times X once it's not. Let's use the first point as the reference points awhile. Minus zero is a goto slope is a difference between wind divided by their fist between excess or 70 minus zero. Go out by 25 months, 50 times x minus 50 from a species. That question off the line is done. Five y close 14 x physical to 14 times 50. And since we're trying to write the question off the words w as a function of why we would like to leave ex alone and write this one as the next physical to 50 minus five watt over 14. Okay, now, the key thing here, as we said before, is that this shape with suspected his work a y axis is Semitic. And this ex, actually, is the question for one health sort of total Wait w would be two times this. Okay, So w would be two times eggs. An area would be w times d one. Okay, Now we found an expression for the area as a functional Find out. Next thing is to do find an expression for that is a functional fought, uh, if Desai reference, if this is the upper service or surface of water and as we go down to pressure will increase. And don't forget what matters is the vertical distance. However, we assume that the are recorded excesses sitting on this in client surface, meaning that if this is if, for us, this is why Bender stepped will be this H institution. Vertical distance. So it means that there's this triangle where this distance is what and distances h and the angle is 30 here. And we would like to find an expression for age as a functional. Why? This is 90 using this triangle be easily see that H is a functional flies equals eight is equal to skirt or three y over, too. No, you know that that is a function of why we know the area's a functional for enemy already know Rocchi. So let's right total force as integral. Why are we integrating again? Because, as I said before, the with as a function of step is changing and the limit as since this is the origin and since this is wise, your level and it since this is wise 70 feet level inter gullible be between zero and 70 Roaches given that is 62.5 D that has a functional quietus Skurdal three oil too in areas of functional boys two times x times. Do you are or be found? X here. So that'll be two times 50 minus five boy over 14 times d y now twos will cancel out. We can write f as less they count all the constants outside. In trickle, we have skirted all three times 62.5 bullets black by in Tegel, 0 to 70. Why? Times 50 minus 55 over 14. C'mon, he feel led this integral and multiplied with screwed up three times 62.5 p. C. That the total hydrostatic force acting all this object the stamp is 8.8 10 36 parents.