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Problem 24

Use the formula cos $C=\frac{a^{2}+b^{2}-c^{2}}{2…

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Problem 23

A walking trail is laid out in the shape of a triangle. The lengths of the three paths that make up the trail are $2,500$ meters, $2,000$ meters, and $1,800$ meters. Determine, to the nearest degree, the measure of the greatest angle of the trail.

Answer

About $82^{\circ}$


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Video Transcript

in this problem, we have a walking trail in the shape of a triangle were given the lengths of each side of the trail. 2500 meters, 2000 meters and 1800 meters. And our goal is to find the measure of the largest angle. So I'm going to label label the angles with A B and C capital letters to label The Vergis is of a triangle and basically we remember from geometry that the largest angle is across from the longest side, so we know the largest angle is angle A. So our goal is to find the measure of angle, eh? And were asked to round to the nearest whole degree. I'm also going to label the sides of the triangle across from angle a side a across from angle. Be aside B and across from angle, see a side seat. We have a formula that helps us find the measure of an angle if we know all three sides. And that is the law of co signs on the Law of Co Sign says that a squared equals B squared plus C squared minus two bc times a co sign of angle, eh, There are other forms of the law of co signs as well. But that's the one that we will want to use, given the information that we have and what we're looking for. So from here we can plug the numbers in 2500 for a and 2004 B and 1800 foresee and keep substituting those numbers in throughout the rest of the formula. And you'll notice that the only thing we have left the only unknown left in the equation is angle, eh? Okay, so now we'll use our calculator. And what I'm going to do is I'm going to take 2500 squared and subtract 2000 squared and subtract 1800 squared. And that gives me negative 990,000 and still on the other side of the equation. We have negative two times 2000 times 1800 that is negative. 7,200,000 times the coastline of angle, eh? The next thing we would want to do is divide both sides by negative 7,200,000 and that gives us 0.1375 You could leave it as a fraction or change it to a decimal. As long as you haven't rounded anything, it's still exact. And finally, to isolate a we need to take the inverse coastline of both sides to undo the coastline function. So we get angle. A is the inverse co sign of 0.1375 That's our exact answer, but we were asked for the answer in degrees rounded to the nearest hole. Never. So you grab your calculator, you make sure it's set two degrees, and you type that in and you get approximately 82 degrees. So that tells us for this walking trail, the largest angle is 82 degrees.

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