An engineer wants to measure the distance to cross a river. If B = 30 ° B=30\degree , a = 300 f t a=300\operatorname{\mathrm{ft}} , C = 100 ° C=100\degree find the shortest distance (in ft \operatorname{ft} ) you’d have to travel to cross the river.

Hey, everyone. So in this practice problem, there's an engineer trying to find the distance to cross a river by using some triangulation here. Let's go ahead and take a look. So we have this, surveyor or engineer who's trying to use some angles and distances to sort of create a little triangle and figure out what's the distance to cross a river. We're told some of the information b little big b equals 30 degrees, a equals 300 feet, and big c equals a 100 degrees. So we're trying to figure out the shortest distance in feet that you would have to travel to cross the river. So what does that mean in terms of our triangle here? Well, let's label the other sides and sort of figure this out. We've got, b, little a, and c already, which means that this is angle a. It's the only one that's left. And that means that, this side over here is little c, because it's opposite of big c, and this side over here is little b because it's opposite of big b. So the shortest distance to cross the river is actually going to be this side over here, the sort of straight path that goes from shore to shore. You wouldn't wanna travel this distance because that's sort of like a longer diagonal distance. So, basically, what this problem is trying to get you to realize is that you're really trying to solve for this side b over here. Alright? Now how do we do that? Well, basically, we're just gonna go ahead and stick to the steps for solving these types of triangles. And what we have here is we have an angle side angle triangle. This is an ASA triangle, not that we had to classify it, but then it just helps us identify what it is, and we can go ahead and use the steps here. So the first one is to draw the triangle and label the given information, but already all that's there. So we're just gonna skip skip to the 2nd step here, which is where you can figure out the 3rd missing angle in our triangle. We already have 2 of them. So how do we figure out this one? We just use the angle sum formula. We have that a plus b plus c is equal to a 180 degrees. So in other words, a is just equal to 180 minus the other two angles. So we have minus 100, minus 30, and what you see here is that a is equal to 50 degrees. So this angle over here is 50 degrees. That's the last missing angle in our triangle. Why is that helpful? Because now we're gonna take this 50 degrees over here, and this is step 2. Now we're gonna sort of use the law of sines to solve for a missing or remaining side. In this case, we only really care about side b. Alright? So the reason we need this a equals 50 degrees is because now we're gonna we're gonna set up our law of sines in step 3. We have sine of big a over a is equal to sine of big b over b, and this is equal to sine of big c over c. Let's take a look here. We have all of the angles. Right? We have all the capital letters here, and then the old lowercase letter I know is a, little a. And then what happens here is I'm actually looking for little b. So basically, when it comes up to set up my equations, remember I need 2 ratios with 3 out of 4 variables. I can't pick these 2 because that's too many unknowns. I have to pick the first two to solve for this little b. Alright? Now we've done this a bunch of times before. I can take this fraction, and because I'm trying to solve for a denominator, I can just flip it upside down and sort of switch the places because it makes it a little bit easier. So in other words, we have that b divided by sine b is equal to a divided by the sine of big a. So I'm just gonna go ahead and plug in some numbers here. Once you cross multiply this to the other side, this ends up being that b is equal to the sign of big b. Big b is equal to 30 degrees, and you're gonna times or you're gonna multiply, and then this is gonna be, little a, which is equal to 300 feet over here, divided by the sign of big a, which is gonna be 50 degrees, which we just found out in the previous step. If you work all of this out, what you should get is a little b distance of a 195.8, and that's gonna be in feet. Alright? So that's basically the shortest distance that you would that that it takes to cross this sort of river or channel here. This b is equal to a 195.8.