A hunter climbs on a tree of unknown height and aims to shoot an antelope at an angle of depression [tex]$38^{\circ}$[/tex]. Given that the horizontal distance of the hunter from the antelope is [tex]$1,500 m$[/tex], calculate the height of the tree, correct to the nearest whole number. [Hint: [tex]$\tan 38^{\circ}=0.7813, \quad \operatorname{Cos} 38^{\circ}=0.7880$, $\sin 38^{\circ}=0.6157[/tex]]
Answer (2)
The height of the tree, calculated using the angle of depression and the horizontal distance to the antelope, is approximately 1172 meters. This was determined using the tangent function and the given angle of 38 degrees. The final answer is rounded to the nearest whole number.
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Define the height of the tree as h and use the given angle of depression 3 8 ∘ and horizontal distance 1500 m .
Apply the tangent function: tan ( 3 8 ∘ ) = 1500 h .
Substitute the given value tan ( 3 8 ∘ ) = 0.7813 and solve for h : h = 1500 × 0.7813 = 1171.95 m .
Round the result to the nearest whole number: 1172 meters.
Explanation
Problem Analysis Let's analyze the problem. We have a hunter on top of a tree, looking down at an antelope. The angle of depression is 3 8 ∘ , and the horizontal distance from the hunter to the antelope is 1500 meters. We need to find the height of the tree.
Define variables and visualize the problem Let h be the height of the tree. The angle of depression is the angle between the horizontal line from the hunter's eye and the line of sight to the antelope. This forms a right triangle, where the height of the tree is the opposite side, and the horizontal distance is the adjacent side to the angle of depression.
Apply the tangent function We can use the tangent function to relate the angle of depression, the height of the tree, and the horizontal distance. We have tan ( 3 8 ∘ ) = adjacent opposite = 1500 h .
Solve for the height We are given that tan ( 3 8 ∘ ) = 0.7813 . So, we have 0.7813 = 1500 h . To find the height h , we multiply both sides of the equation by 1500 : h = 1500 × 0.7813 = 1171.95 meters.
Round to the nearest whole number We need to round the height to the nearest whole number. Since 1171.95 is closer to 1172 than 1171 , we round up to 1172 meters.
Final Answer Therefore, the height of the tree is approximately 1172 meters.
Examples
Imagine you're designing a zip line from a tree to a platform on the ground. Knowing the angle of depression and the horizontal distance, you can calculate the height of the tree needed to set up the zip line safely. This is a practical application of trigonometry in recreational engineering.
