Make sure your calculator is in degrees mode for this question. Suppose an observer (viewer 1) hascalculated that from a point 30 metres from (the middle of) the base of a tree on flat terrain, the angleof their line of sight to the top of a tree is 25◦ , as shown in the diagram. The height of the tree is hmetres, and the viewer’s eye level is 1.8 metres above the ground.Perform calculations using at least 4 decimal places, round off your final answers to 2 decimal places.Show your working.(a) Use trigonometric functions to calculate h.(b) Viewer 2 is at a position 15 metres from the base of the tree and their eye level is 1.5 metresabove the ground. Calculate the angle of the line of sight of viewer 2 to the top of the tree
Question
Make sure your calculator is in degrees mode for this question. Suppose an observer (viewer 1) hascalculated that from a point 30 metres from (the middle of) the base of a tree on flat terrain, the angleof their line of sight to the top of a tree is 25◦ , as shown in the diagram. The height of the tree is hmetres, and the viewer’s eye level is 1.8 metres above the ground.Perform calculations using at least 4 decimal places, round off your final answers to 2 decimal places.Show your working.(a) Use trigonometric functions to calculate h.(b) Viewer 2 is at a position 15 metres from the base of the tree and their eye level is 1.5 metresabove the ground. Calculate the angle of the line of sight of viewer 2 to the top of the tree
Solution
(a) To calculate the height of the tree, we can use the tangent of the angle of elevation, which is the angle formed by the line of sight and the horizontal. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
In this case, the angle of elevation is 25 degrees, the adjacent side is 30 meters (the distance from the observer to the tree), and the opposite side is the height of the tree (h) minus the height of the observer (1.8 meters).
So, we have:
tan(25) = (h - 1.8) / 30
Solving for h gives:
h = 30 * tan(25) + 1.8
Using a calculator, we find that tan(25) is approximately 0.4663. So:
h = 30 * 0.4663 + 1.8 = 16.789 + 1.8 = 18.589 meters
So, the height of the tree is approximately 18.59 meters, to two decimal places.
(b) To find the angle of the line of sight of viewer 2 to the top of the tree, we can use the inverse tangent function (also known as the arctangent), which gives the angle whose tangent is a given number.
In this case, the opposite side is the height of the tree (18.59 meters) minus the height of the observer (1.5 meters), and the adjacent side is 15 meters (the distance from the observer to the tree).
So, we have:
angle = arctan((h - 1.5) / 15)
Substituting the value of h gives:
angle = arctan((18.59 - 1.5) / 15)
Using a calculator, we find that this is approximately 48.37 degrees.
So, the angle of the line of sight of viewer 2 to the top of the tree is approximately 48.37 degrees, to two decimal places.
Similar Questions
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