Observers at positions A and B 2 km apart simultaneously measu...

Question

Observers at positions A and B 2 km apart simultaneously measure the angle of elevation of a weather balloon to be 40° and 70°, respectively. If the balloon is directly above a point on the line segment between A and B, find the height of the balloon.

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Two observers are located at points C and D, which are 3 kilometers apart along a straight line. They simultaneously measure the angle of elevation of a drone flying above the line segment CD. Observer C measures an angle of elevation to B 30 degrees while observer D measures the angle of elevation to B 60 degrees. Assuming that the drone is directly above the line segment between C and D, what will be the height of the drone located above the ground. Awesome, so it appears for this particular prompt, the final answer that we're ultimately trying to solve for is we're trying to figure out the height of the drone that is located above the ground. Awesome, so assuming that the drone is flying directly above the line segment between C and D. Awesome. So first off, it's important to note now that we know that we're trying to solve for the height of this drone, that it's going to be, we're gonna have to use like geo geometric identities, so we're gonna be dealing with like geometry and trig identities to help us solve for this particular problem. So as we can see. In a moment we're going to be using a triangle to help us solve for this problem. So with that said, let's take a moment here to read off our multiple choice answers to see what our final answer might be, noting that they're all in the same units of kilometers. So A is 2 multiplied by the square root of 3 divided by 3. B is 3 multiplied by the square root of 3 divided by 4. C is 3 multiplied by the square rooted 2 divided by 4, and D is 3 multiplied by the square rooted 2 divided by 2. Fantastic. So first off, let us denote that the value H denote the height of our drone, and ultimately that's our final answer that we're trying to solve for. So H equals some value that we don't know, and that's going to be the height of our drone. So now let's take a moment to help us draw a diagram to help us better visualize this problem to see what exactly is happening. So as you can see, like I said before, we're going to be dealing with a triangle to help us solve for this particular problem. So as you can see, we have our dotted line which denotes the height, and let's say we have our drone, which I'll draw a blue star to represent our drone. So we have our drone that's flying at some distance H above the ground. And we have two observers that are standing on the ground, which I'll draw two little stick figures that have two different perspectives. So we have. Two different So like, so we have two different perspectives. So as we can see, we have our two different perspectives. From the 30 degree angle of elevation, and we also have our 60 degree angle of elevation. So, once again, we have two different observers. So we have an observer on the left side, and we have an observer on the right side. And as you can see on the observer on the right side sees an angle of elevation to be 60 degrees, whereas our observer on the left sees an angle of elevation to be 30 degrees, and this is when they're standing on the ground looking up at the drone. So both of our observers are standing on the ground looking up at the drone. So, from where our observer is on the left to where the where the drone is flying some distance H above the ground is going to be some distance X, and the distance between both observers is a distance of 3 kilometers apart. So that will mean that our observer on the right, going to where the height of the drone is. So from where the height of the drone is to where the observer on the right is standing is going to be some distance 3 minus X. Awesome. So, with that said, using our trig slash geometric identities, we should be able to state and recall that tangent of 30 degrees. So we need to note that tangent of 30 degrees. is going to be equal to H divided by X, and then therefore we can say that H is equal to X multiplied by tangent of 30 degrees when we rearrange our expression to isolate and solve for H all by itself, which once again H is the height of the drone above the ground. And similarly, for a tangent of 60 degrees, it's going to be equal to H divided by 3 minus X, considering our different legs of the triangle because we're dealing with right triangles. Awesome, as you can see. So with that said, we can now rearrange our tangent of 60 degrees expression to isolate and solve for H. and when we do, we'll find that H is equal to 3 minus X multiplied by a tangent of 60 degrees. Fantastic. So now we need to equate the two expressions that we found for H, so we need to say that X multiplied by tangent of 30 degrees. is going to be equal to drum roll, 3 minus X multiplied by tangent of 60. So 3. Minus X multiplied by a tangent of 60 degrees. So now what we need to do is we're trying to rearrange this expression to isolate and solve for X all by itself. So I'm going to skip a few steps, but ultimately, when you've used your algebra correctly to rearrange this expression to isolate and solve for X, you will find that X is equal to 3, multiplied by the tangent of. 60 degrees divided by the tangent of 30 degrees plus the tangent of 60 degrees. Fantastic. We also need to recall once again that H is equal to X multiplied by tangent of 30, so we can therefore substitute our value of X is equal to 3 multiplied by the tangent of 60, divided by the tangent of 30 degrees. Plus the tangent of 60 degrees. So when we do that, we will find that when we substitute in that value of X, we'll find that H is equal to 3 multiplied by the tangent of 60 degrees, divided by the tangent of 30 degrees, plus the tangent of drum roll, 60 degrees, multiplied by the tangent of 30 degrees. And when we perform, This math for our two different perspectives, we will find that. The tangent of 60 degrees is going to be equal to sine of 60 degrees divided by the cosine of 60 degrees. So the sign of 60 degrees divided by the cosine of 60 degrees, which is going to be equal to the square root of 3 divided by 2. So the square root of 3 divided by 2. All divided by 1 divided by 2, which is going to be equal to simply the square root of 3, whereas from our other observation, which is the tangent of 30 degrees, which is going to be equal to sine of 30 degrees divided by the cosine of 30 degrees, which is going to be equal to 12 divided by the square root of 3 divided by 2, which is going to be equal to. The square root of 3 divided by 3. So, therefore, without a doubt, we can go ahead and write that. H, the height is going to be equal to parentheses 3 multiplied by the square root of 3, divided by the square root of 3. So we need to take 3 multiplied by the square root of 3, divided by the square root of 3 divided by 3 plus 3, all multiplied by the square root of 3 divided by 3. So when we simplify down this expression down to its most simple form, we will find that H is simply equal to 3 multiplied by the square root of 3. Divided by 4, and its units of measurement are going to be in units of kilometers. And that is our final answer. Hooray, we did it. So, once again, our final answer, without a doubt, looking at our multiple choice answers. To be multiple choice answer letter B, which once again is 3 multiplied by the square root of 3, divided by 4 kilometers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Observers at positions A and B 2 km apart simultaneously measure the angle of elevation of a weather balloon to be 40° and 70°, respectively. If the balloon is directly above a point on the line segment between A and B, find the height of the balloon.

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