Alice wants to take a picture of a historical building that is wide but not tall. From available references, she is able to find out that the building is 33 metres wide. She plans to take the picture standing directly in front of it, at a point perpendicular to the midpoint of the building. She is using a moderately wide-angle lens which provides an angle of coverage of 64◦. Alice sketches the situation as in Figure 7, where all lengths are measured in metres. The point A represents Alice’s position. Points B and C are at each end of the building. Point D is the midpoint of the building. The length of AB is equal to the length of AC. The angle BAD is 32◦. How far from the midpoint D should Alice stand in order to be able to take a picture of the whole building? (You can assume that the height of the building is not an issue)
Question
Alice wants to take a picture of a historical building that is wide but not tall. From available references, she is able to find out that the building is 33 metres wide. She plans to take the picture standing directly in front of it, at a point perpendicular to the midpoint of the building. She is using a moderately wide-angle lens which provides an angle of coverage of 64◦. Alice sketches the situation as in Figure 7, where all lengths are measured in metres. The point A represents Alice’s position. Points B and C are at each end of the building. Point D is the midpoint of the building. The length of AB is equal to the length of AC. The angle BAD is 32◦. How far from the midpoint D should Alice stand in order to be able to take a picture of the whole building? (You can assume that the height of the building is not an issue)
Solution
To solve this problem, we can use the properties of a right triangle and trigonometry.
The situation described forms a right triangle where AD is the adjacent side, AB is the hypotenuse, and the angle BAD is 32 degrees. We want to find the length of AD.
We know that the cosine of an angle in a right triangle is equal to the adjacent side divided by the hypotenuse. So, we can write the equation:
cos(32) = AD / AB
We also know that the building is 33 meters wide, so the length of BD (half the width of the building) is 16.5 meters.
Since AB = AD + BD, we can substitute into the equation:
cos(32) = AD / (AD + 16.5)
To solve for AD, we can rearrange the equation:
AD = 16.5 * cos(32) / (1 - cos(32))
Using a calculator, we find that cos(32) is approximately 0.8480480961. Substituting this value into the equation gives:
AD = 16.5 * 0.8480480961 / (1 - 0.8480480961)
Solving this equation gives AD = approximately 96.5 meters.
So, Alice should stand approximately 96.5 meters from the midpoint of the building to be able to take a picture of the whole building.
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