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Numerade Educator

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Problem 4 Medium Difficulty

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Answer

8333.33 $\mathrm{lb}$

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Video Transcript

in this problem were given. Others were trying a lettuce summers water and we rest approximated force acting on one side off this triangle we know that force is equal to pressure Times area We can like this one as role times G times d ward A step months fly by area and we can else lark Just want a specific wage Times depth claims area We know what specific wait is that a £62.5 per feet cubed for water. So this part is known. We need to determine depth in the area. Let's start from area area is the area off this thin, strip shaded region in decorated along this edge. So we would then if he's women, we need to approximate the area off this. And if you assume that this is an infinitesimal thickness d x d area would be done D x times, whatever dis ling l. A is now in, it'd down this link. In order to do that, he first need to find the question off this line. So let's assume not, um, we have an origin located right at this point that the stars, you're zero point this point would be or this level would be executed to end this level. But B X equals seven. Um, deaths would be y equals zero, and Destin would be like a stand. So in order for that land, accordance off that line would be to zero and seven 10 in on any question off the line will be up for y minus one on a slope Times X minus X not, uh s slow. Using this given Florence can do count as 10 minus zero divide by seven minus two. The difference between winds divide by difference between excess that is 10 over five sort of slope is too. We could easy drop this points. Um, let's use this first point. So let's write this one is why minus zero is equal to two times explains to. So why's than equal to two x minus? Or if that is the case, we can done right this area as, uh since this is why in terms of exes to explain us for it means that link is to X minus force, or the area would then be to ex ones for us by by T X, and this would be actually this infant decimal area off this tin strip. So in order to find a total area of you would need to integrate integrated, or we would need to some of those areas up. Now the next thing to determine is deed that so deep Ah, let's say that we're measuring this from the surface. And since these are X levels less than that that as ex that also changes depending on where art in strip at, so we can then write the force or infant Izabal Force Acting oldest infanticidal strip as, um 62.5, which is Delta months fly by x Times two X minus 40 x. Now we would need to integrate it. We can also write this one as a remote. Some sort of filler force would be limit as I negotiate Infinity summation off those old 10 strips from one end 62.5 My life by X times to explains for the X, or we can write this oneness Delta eggs and we can actually work that intern in trickle or the limbs off the interval will be the limits off X. We signed that exchanges between two and seven for this given right triangles, allegedly from 2 to 7 62.5 times x times two x minus four D x That is 62.5 is a constant 227 to expert months for its D X. That will be 62.5 is again just constant. He had two X Cube or three minus two. Expert community elected two and seven. If they do so, we find out the force acting on the size of the strike triangle as 8333.33 pounds.