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Question Video: Finding the Angle of Elevation Mathematics

The distance between two buildings is 40 m. The top of building ๐ถ๐ท has an angle of elevation of 30ยฐ, measured from the top of building ๐ด๐ต. If the height of building ๐ด๐ต = 30 m and the bases of the two buildings are on the same horizontal plane, then the height of ๐ถ๐ท to the nearest meter = ๏ผฟ m.

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Video Transcript

The distance between two buildings is 40 meters. The top of building ๐ถ๐ท has an angle of elevation of 30 degrees, measured from the top of building ๐ด๐ต. If the height of building ๐ด๐ต equals 30 meters and the bases of the two buildings are on the same horizontal plane, then the height of ๐ถ๐ท to the nearest meter equals how many meters.

So weโ€™ve been given the diagram to go alongside this problem. And all the information in the question has been labeled on it. We have an angle of elevation here of 30 degrees formed between the horizontal and the line of sight as we look up from building ๐ด๐ต to building ๐ถ๐ท. We also know the height of building ๐ด๐ต, itโ€™s 30 meters, and the horizontal distance between the two buildings, 40 meters. Weโ€™re told that the two buildings are on the same horizontal plane, which simply means we can assume that the ground between them is flat.

What weโ€™re looking to calculate is the height of building ๐ถ๐ท. From the diagram, we can see that this will be composed of two lengths, a portion which is the same height as building ๐ด๐ต, so thatโ€™s 30 meters, and a portion which is this currently unknown length here, which we can think of as ๐‘ฅ meters. We can also see that this length ๐‘ฅ is one side in a right triangle. And in this triangle, we know one other side of 40 meters and one angle of 30 degrees. We can therefore apply right-angle trigonometry to calculate ๐‘ฅ.

Labeling the sides of this triangle in relation to the 30-degree angle, we can see that we know the adjacent and we want to calculate the opposite. So itโ€™s the tan ratio that weโ€™re going to use. Recalling that tan is opposite over adjacent, we have that tan of 30 degrees is equal to ๐‘ฅ over 40. Multiplying both sides of this equation by 40, we have that ๐‘ฅ is equal to 40 multiplied by tan of 30 degrees. And evaluating this on a calculator, ensuring our calculator is in degree mode, we find that ๐‘ฅ is equal to 23.0940 continuing.

We havenโ€™t quite finished though because we need the total height of building ๐ถ๐ท. So we need to add on the additional length of 30 meters. Doing so gives 53.0940. Weโ€™re asked to give our answer to the nearest meter. So as the digit in the first decimal place is a zero, we round to 53. So by applying trigonometry in the right triangle formed by the horizontal, the vertical, and the line of sight, we found that the height of building ๐ถ๐ท to the nearest meter is 53 meter.

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