when-two-railroad-tracks-merge-the-overlapping-portions-of-the-tracks-are-in-the-shapes-of-circular-

Question

When  two  railroad  tracks merge,  the  overlapping  portions  of  the  tracks  are  in  the shapes  of  circular  arcs  (see  figure).  The  radius  of  each arc $ r $  (in  feet)  and  the  angle $ 	heta $ are  related  by 

$ \dfrac{x}{2} = 2r \sin^2 \dfrac{	heta}{2} $

Write  a  formula  for $ x $ in  terms  of $ \cos 	heta $.

Sheryl Ezze · Accepted Answer

So we&#x27;re given this formula X over to is equal to two are sine squared of theta over two and we want to change that. So X is a function of a co sign. So we wanna have X and co sign in the function, which means we don&#x27;t want this angle here, and it didn&#x27;t even say and co sign of data we just wanted co sign. So we have to get rid of the sine squared. So we have an identity that says that if you have the sine squared oven angle, they give a formula in terms of you that that that&#x27;s equivalent to one minus the co sign off to you all over to. And this comes from actually the the double angle formula for co sign and doing a little bit of substituting. So this is the identity that we want to use. So let&#x27;s do it. So we have I will do a little substitution. We have X over to which will multiply both sides by two in a minute, is equal to to our times. And now we needed to do a substitution for sine squared. And so we&#x27;re gonna have one minus and you is acting as data over to So I have one minus cosine off. And then I need to take two times you and this whole thing is acting as you. So I have data over to and nicely enough we can see what&#x27;s going to end up happening there, that these twos, they&#x27;re going to cancel and lo and behold, these twos they&#x27;re going to cancel. And then if I multiply both sides by two, will have X is equal to, and I have a two are here. So this combines together to be to our and I&#x27;m going to distribute it. So I have to our times one minus to our times the co signed of data and there is our function written in terms of cosign, and that&#x27;s a simpler function to plug things into, then is this function and we&#x27;re all done

Ankit Gupta · Answer

it is given Exabyte two equal toe are science is quiet. Double the question that idea formula for Xingdong Costinha. So executed. This is for the poor. Uh, sign is what people so poor are Sinus. 40 days Seem it one minus course, Tita. Way to visit A horrible Remember that sine squared away toe one minus caused it away is a formula. So boots were full. You are Look at one minus costs, Peter. So this is answer big.

Himanshu Kushwaha · Answer

figure like this in which we have Did you two are the radius R R. This is gold and this angle s tutor divide by two decide from here to here is given it be and decide from here to here is given as 50. Now we have to find the value off course. Chatur divide by two in part a now course tutor divide by one to base divided by hyper tennis. It means we have to find the length off this side. This wall side is quarto are because this is the radius. So this will be 1/4 toe ar minus B. It means base is a Puerto R minus B and high. Partain IUs is equal to our support. This is Oh, this is a This is B here, refined that it&#x27;s equal to r minus. B and O B is equal to our using the do re friend of a look. Question Divide by two In second part, we have to find the value off. 10 Tater divide by four Viewed one minus course tutor Divide by two divided by scientists are divided by two formula and in the case sign Tita divided by two will be equal toe perpendicular. It means be divided by high protein statements. All be This gives us 50 divided by are viewed this value up. Sign to Daddy right about no. One minus R minus. Be divided by r Divided by 50. Divided by Are we take the Elsom inhuman? It&#x27;ll begin ar minus R plus b divided by r divided by 50. Divided by r r Cancel our and this our consultants are bigger. This is equal to be divided by 50 This is the value off 10 teeter divide by four

Anjali Kurse · Answer

Hey. So We have one second to pull up this problem. The circular arc of a railroad has a curve has a cord of like 3000 ft corresponding to a central angle of 40°. So let&#x27;s draw that 40° might look something like that. Actually. Let&#x27;s make this circle a little bit bigger so it&#x27;s easier to see. Okay, Say that is a 40° angle. Yeah. And here is our cord of 3000 ft. 1000 ft. Okay, so here&#x27;s our diagram that represents the situation. Show the known quantities on the diagram. Use the variables are and asked to represent the radius of the arc and the length of the ark respectively. Okay, so this is gonna be our the radius of the circle and this part here is going to be us. That is the ark. Now we&#x27;re going to find the radius of the circular arc. Okay, how are we going to do that? Okay, so We have this 40° here. We have this 3000 ft here. What we can do is that since these are both are we know with some geometry properties that these angles at the bottom are going to be identical. Let me redraw that this way. So maybe it&#x27;s a little easier to see. So this is our 3000 sides. This is our 40 degrees. And remember these are the radi I of the circle. So they&#x27;re equal because already I of a circle are equal. I know the circle isn&#x27;t perfect but all of these are our they&#x27;re all going to be equal to each other. Okay, so if these sides are equal with that property of triangles, these are going to be equal these ankles. So if we have that these angles are equal. Well, we have the third. So let&#x27;s call these angle A. So a plus a. plus 40 Is going to be equal to 180° in the triangle total. So to a It&#x27;s going to be equal to 140 A. is 70°. Cool. Cool. Cool, 70° 70 degrees. Okay, what&#x27;s the next step? Well, we have to find our right, we also have to find S. S. Is the last piece. So let&#x27;s focus on part B. Which is finding our Okay, how do we find our? Well, check this out. We have 40° corresponds to 3000 And a. Which we know now which is 70° corresponds to an unknown. If you&#x27;re thinking a lot of science, you are totally right. So signed 40° over 3000 Should be equal to sine 70° over our. Okay, so let&#x27;s just simplify the left hand side of that before we go too far because that gets confusing. Make sure your calculator is a degree mode because we&#x27;re using degrees for this. So then when we saw signed 40 over 3000, it&#x27;s gonna be a little ugly. Oh, that&#x27;s a very small number. 0.000214. Okay, forget that step. We&#x27;re actually not gonna skip that because that&#x27;s too weird. We are going to solve for R. Here. So what can we do? Well, we can use the rule of proportions here. Multiply this by this and this by this. So are signed 40° should be equal two, sign 70 degrees. So to isolate the R. We just divide this 3000 signed 70° over sign 40°. So let&#x27;s plug that into our calculator, 3000 signed 70 over signed 40. And we get Are equal to 4385.71. Okay, there we go. That&#x27;s our now. The third step which is find S. So S. Is the length of the circular arc. Right this part here. So it&#x27;s a piece of the circumference of the circle. So let&#x27;s find the circumference of the circle two Pi R. That&#x27;s how you find it. And we have our we solve for R. Appear. So let&#x27;s just put that into our calculators two times pi Temps, and we get 27,500 56. Or if we want to keep that in terms of pie, all we have to do is multiply it by two and then keep the pie on the outside. But I&#x27;m not going to do that. Okay, the next step is to find the ratio, see how this is a ratio of the circumference. It&#x27;s not the full circumference, it&#x27;s just a piece of it. But it corresponds to this 40°. So what we want to do here is set up a ratio So 40° of the circle which has a total of 360. Let&#x27;s write this down 40° out of 360. Should be equal to whatever S. is whatever S. Is over the full circumference which we have. Okay .207- seven. Now we&#x27;re going to solve this the same way that we solve the other. But actually we can simplify this. What&#x27;s 4/36? We can cross out the zeros here, 1/9 just to make it a little bit easier. S over the same number. 27,000 556.207- seven etc etc. Okay, now we&#x27;re going to cross multiply nine S. Equals 27556.207- seven. We divide that by 9 to isolate that. S and we get 3061.8 okay, so to recap first we drew this accord is just a line segment across the circle. The longest chord of a circle is the diameter which would go straight across. But this chord is like 3000 ft corresponding to an angle angle of 40°. The next step is finding our we were able to figure out that these angles are the same because the sides of the ink, the sides of the triangle are the same, therefore the angles are the same. That&#x27;s back from geometry. We solved and we found that those angles are 70° each. So then we use the law of sines to find our our because we have 40° across from 3000, so we can use 70° across from our. We saw we found R equals 4385.71. Our final step was to find S. So we found the proportion of the full circle that this angle response to and S. Is proportional to that. So we set up disproportion this ratio here, we solved for us, and we got 3,061.8.

Antrell Banes · Answer

Hello. Um, so our problem for today says, Ah, the circle of arc of a railroad curve has a quart of left 3000 feet and a central angle 40 degrees. The problem was just a dry diagram. The represents the problem and then labeled that dark ground with known quantities and then labor the radius are and the r Cliff s. Then the problem was just defined that argument and that radius. So to begin this problem, we&#x27;re gonna draw the diagram. It&#x27;s this has said that it was a circle of curve. We&#x27;re just gonna first a dry circle, okay? And we&#x27;re going to draw the central angle for that circle. So we first need to draw the center. And the central legal is just an angle that ah, who&#x27;s vortexes on the center. And so this will be our central angle. Just 40 degrees and then another known Quincy is the court of that art, which is 3000 feet in the court Court is just a live statement that goes from one point on the circle to another point in the circle, and that lift is 3000 feet. Okay, so when the unknowns is the radius, which is just a line segment from the center of the Circle two. Any point on the circles of this life of represent Radius? And then, of course, the other known is Thea are Cliff Ah! And that our Clint will be this curve you, and that&#x27;s going to be s Okay, So now we have our diagram. The first thing we gotta do is find our radius because, um, in order, find the art life. We have to find our radius first. So the method gonna use to find the radius is a right triangle. Trigonometry. So if we were to draw that triangle that&#x27;s on the inside of the circle on the outside of the circle and then we&#x27;re going to split that that triangle into two equal parts will see that we have to right triangles. Okay, so one of those right triangles, we just look at one right triangle. We could see that the length of this side of the triangle will be half of the 3000 feet. That was the whole entire length of this triangle. So this will be 1500 feet way. We can also see that the this angle here this angle would be 20 degrees because it will be half of the 40 degrees that was here. And then, of course, this was in the circle. This line here was the radius, and this line will still be the radius. But now the radius it represents the have hot news of our new right triangle. So if it says we have a right triangle Ah, we can use Ah, right triangle trigonometry if we&#x27;re given the opposite. So this is our reference angle here. This was our represent goal. That was our represent. Go. It were given the opposite of that reference angle, which is the 15 100 feet. And we&#x27;re looking for the how part news, which is our radius. Um, Then we can use, um sign, uh, tricking of tricking a metric function sign which is equal to a fine of failure, which is our angle is equal to the opposite over the hot news. So I&#x27;m going to substitute are known values into this equation, so that gives us sign of fatal, which is our 20 degrees in this case is equal to the opposite, which is 1500 over the high partners, which in this case is our Hey, they&#x27;re solving for our we have the use in verse operation. So since ours being divided here, we&#x27;re gonna have to multiply, are on both sides. Thes two hours would cancel out, leaving us with our sign. 20 degrees equals two 1500. Ah, Using inverse operations again, we will now have to divide both sides by sign 20 degrees. And these two signed 20 degrees would castle out leaving What&#x27;s what are is equal to 1500 divided by side, 20 degrees. And if we use the calculator, we&#x27;ll see that 1500 divided by signed 20 degrees. It&#x27;s 4000 385 0.71 feet and that is our radius. Okay, so the next thing we have to do for this problem is fine. The are clip and our Clint equation, the formula for our Clint, this case s equals two pi r times data provided by 360 degrees. And so if you notice that two pi r is just the circumference of the whole entire circle, but we don&#x27;t eat the whole the length of the whole entire circle. We just need part of that. And what part do we need. We need the argument s, which is a fraction of the whole entire circumference. On what fraction is that? Well, this is just the 40 degrees out of the whole entire 360 degrees. So that&#x27;s where this form the argument formula came from. Um, so, substituting all our values that are known values into the formula we get to terms. Pie ties are which is radius, which we just found, which is 4300 85 0.71 times fatal, which is our central angle of our original um, circle, which is four degrees divided by 360 degrees. If we put that in our calculators, we&#x27;ll see that. That gives us 3000 61 point 81 I&#x27;m sorry. 80 and that&#x27;s our Argh. Let s So we found our radius and our argument. All right, That&#x27;s it.

Zulfiqar Ali · Answer

on the given day job. We have angle feta. Well, Peter equals five degrees and then we have radius. Well, radius is equal to 500 meter and x relation due to gravity. That is 9.8 meters per second squared. And we need to find out speed. Well, the velocity or the speed is given by damn pita guaranteed A is equal to the square divided by R times achy. Now let&#x27;s hope for free. Then we is equal to root are times g times Dan Peter! Damn pita. Let&#x27;s call the secret to number one Now we do substitution in equation number one We have re equals your Route 500 Multiply by 9.8 times Stampeder. Well, Peter is, uh, five degrees and solving this velocity equals 20.7 meters per second.

Sheryl Ezze · Answer

okay. Were asked to write X uh, this expression in terms of coastline data. You have excellent itude that he could have two times are signed. Great. Big trouble, too. Okay. You know our sign for me, which is a plus minus one minus. Both signed data. We&#x27;ll get you part two. That&#x27;s actually what you wish you could too are I am one minutes coast lying there over to choose. Canceled. Well, up with X. Over to you. To our ways, Marco. Sign there and multiplying by two on the outside. Get expert. Secretive too. Are that is to our coastline.

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The mach number $ M $ of an airplane is the …

Problem 136 Hard Difficulty

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$x=2 r-2 r \cos \theta$

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