Solve each problem. See Examples 3 and 4. Height of an Antenna...

Question

Solve each problem. See Examples 3 and 4. Height of an Antenna A scanner antenna is on top of the center of a house. The angle of elevation from a point 28.0 m from the center of the house to the top of the antenna is 27°10', and the angle of elevation to the bottom of the antenna is 18°10'. Find the height of the antenna.

Accepted Answer

Hey, everyone in this problem, we're asked to find the height of a flagpole positioned at the center of the roof of a building where the angle of elevation from a point that's on the ground. 35 m away from the center of the building to the top and bottom of the flagpole is 40 degrees and 30 minutes and 35 degrees, 20 minutes respectively. We're given five answer choices all in meters. Option A 5.08 option B 8.05 option C 54.70 and option D 24.81. Let's get started by drawing a diagram of what we have going on. OK. We're giving a lot of information. So drawing a diagram is just gonna help us figure out how it all relates. And so we're gonna start with this building. Yes, we have a building and we know that we have a flagpole on the roof at the center of the building. Yeah. So this is our flight pool and it is right in the center of that group and this black pool has some height which we'll call and that height H is what we are looking for and the height of the flight pole. Now, the other information we're told is that we have this 0.35 m away from the center of the building on the ground. And so our flag poles at the center of the roof, so it's gonna be 35 m away from there on the ground. And we have this point and from this point, we have two elevations measured. The first is from the top of the fly pool to our point. And the second is from the bottom of the flight to our point. Ok. So we can draw these lines and we know the elevation. So to the bottom of our flag pole, we have the smaller angle which is 35 degrees, 20 minutes and to the top of our flagpole, we have this larger angle and that's gonna be 40 degrees and 30 minutes. All right. So now that we have this drawn out, what we can see is that these angles of elevation, this distance of 35 m and the height of the building or the flagpole, these make up some nice right triangles that we're gonna be able to use to find the height H we're looking for. Ok. There's a couple of other distances we should define. So the entire distance from the top of the flagpole to the bottom of the building, that's the left hand side of our big triangle. So we're gonna label that and that's gonna be called DT. So that's gonna be the total distance and then we have the height of the building itself which forms the side of our smaller triangle and we're gonna call that DB. Ok. So the distance or the height of the building and what we can see when we label those heights is that the height we're interested in h it's just going to be that total height. DT minus the height of the building DB. OK. So now what we can do is use the triangles we've drawn to find DB and DT that will allow us to find the height. H. OK. OK. So let's start with the smaller triangle. And before we can get started with the smaller triangle, we need to put this angle into degrees. OK? So let's do that. We are going to put our angles into degrees. So let's think about this 20 minutes. We have 35 degrees and 20 minutes. So if we have 20 minutes recall that this is gonna be equivalent to 20 minutes multiplied by one degree per 60 minutes. OK? Every one degree we have 60 minutes. And so our 20 minutes is gonna be equivalent to 0.333 repeated degrees, which tells us that 35 degrees and 20 minutes will just be equal to 35.333 repeated degrees. OK. Similarly, for our other angle and we have this 30 minutes with our other angle. We're gonna take our 30 minutes multiply by one degree per 60 minutes. The unit of minutes divides it and we're just left with 0.5 figures. What that means is that our angle of 40 degrees and 30 minutes is just going to be equal to 40.5 degrees. Yes, we have both of our angles in terms of degrees only which allows us to use our trigonometric functions and our ratios on them. So let's start with finding DB the fight of the bill. Now, if we take a look at the diagram again, the height of the building is the side, a right triangle with angle 35.333 degrees. Mhm It's the opposite side to that angle. And we know that the adjacent side is 35 m. So we're relating the opposite side and the adjacent side, we can do that through tangent right, because we have a 90 degree angle, we can use our trigonometric ratios. And we know that tangent of our angle theta is gonna be equal to the opposite side divided by the adjacent side. So in our case, tangent of that angle of 35.333 degrees is going to be equal to DB, the height of the building divided by 35 m. We can multiply both sides by 35 m. And we get that the height of the building DB is approximately 24.812 m. OK. So we've got the height of the building. Remember, we're trying to find the height of this flagpole. So now we're gonna switch over and we're gonna find the total height. Now we're gonna use the triangle with the larger angle. It's gonna have a side given by that total height. Now again, we're gonna relate it through tangent because the adjacent side is still going to be that 35 m. We've been told the opposite side is that total distance or total height we're looking for. And so the process is gonna be very similar. So refining DT through a tangent and again, tangent of our angle, theta is gonna be the opposite side divided by the adjacent side. In this case, we get tangent of 40.5 degrees is equal to our total distance DT divided by that adjacent side of 35 m. We can multiply both sides by 35 m. And we get that the total height DT is equal to about 29.893 m. Awesome. Once again, going to our diagram, we now have this total height BT, we have the height of the building DB. If we take that total height and subtract the height of the building, we're gonna be left with the height of the flagpole that we want. So the last step is to do a subtraction. And so our height H is going to be equal to 29.893 m minus 24.812 m, which gives a value for H of 5.081 m. If we compare this to the answer choices and we round, what we'll see is that our answer corresponds with option A 5.08 m. Thanks everyone for watching. I hope this video helped you in the next one.

Key Concepts

Angle of Elevation

Right Triangle Trigonometry

Difference of Heights Using Angles

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