Alice had been standing on the ground (Point A) and observing a brightly colored object resembling a bird on the top of a tree at a distance of 4 meters from the tree. She decided to get a closer look by moving 2 meters closer to the tree (Point B). After moving closer, she realized that the object was not a bird but something that she could catch. Then, she decided to catch it by climbing the tree, which had a height of 6 meters from the ground.Using the above scenario, please answer the following questions showing step by step calculations and stating the formulae.(i) Find the angles formed by Alice at the points A and B relative to the top of the tree. What are these angles called as?(ii) Determine whether angle A is larger than angle B. Make a conclusion about the comparison of angles when observing an object from a distance versus close.(iii) Find the distances between the object and points A and B.
Question
Alice had been standing on the ground (Point A) and observing a brightly colored object resembling a bird on the top of a tree at a distance of 4 meters from the tree. She decided to get a closer look by moving 2 meters closer to the tree (Point B). After moving closer, she realized that the object was not a bird but something that she could catch. Then, she decided to catch it by climbing the tree, which had a height of 6 meters from the ground.Using the above scenario, please answer the following questions showing step by step calculations and stating the formulae.(i) Find the angles formed by Alice at the points A and B relative to the top of the tree. What are these angles called as?(ii) Determine whether angle A is larger than angle B. Make a conclusion about the comparison of angles when observing an object from a distance versus close.(iii) Find the distances between the object and points A and B.
Solution
(i) To find the angles formed by Alice at points A and B relative to the top of the tree, we can use the tangent of the angle which is the ratio of the opposite side to the adjacent side in a right triangle.
At point A, the distance to the tree is 4 meters and the height of the tree is 6 meters. So, the tangent of angle A is 6/4 = 1.5. Therefore, angle A = arctan(1.5) = 56.31 degrees.
At point B, the distance to the tree is 2 meters (4-2) and the height of the tree is 6 meters. So, the tangent of angle B is 6/2 = 3. Therefore, angle B = arctan(3) = 71.57 degrees.
These angles are called the angle of elevation.
(ii) From the calculations above, we can see that angle A (56.31 degrees) is smaller than angle B (71.57 degrees). This means that the angle of elevation increases as one gets closer to the object.
(iii) To find the distances between the object and points A and B, we can use the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
At point A, the distance to the object is sqrt(4^2 + 6^2) = sqrt(16 + 36) = sqrt(52) = 7.21 meters.
At point B, the distance to the object is sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40) = 6.32 meters.
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