Coconut palm trees can reach heights of up to 100 feet. Suppose you are lying on the beach at a distance of 60 feet from a 48-foot tall palm tree. What is the angle of elevation from your position to the top of the tree? Round to the nearest tenth if necessary.
a. $128.7^{\circ}$
b. $53.1^{\circ}$
c. $36.9^{\circ}$
d. $38.7^{\circ}$
Answer (1)
Define the angle of elevation as θ and use the tangent function: tan ( θ ) = 60 48 .
Calculate the angle of elevation by taking the inverse tangent: θ = arctan ( 60 48 ) .
Evaluate the arctangent: θ ≈ 38.659 8 ∘ .
Round the angle to the nearest tenth of a degree: 38. 7 ∘ .
Explanation
Problem Analysis Let's analyze the problem. We are given the height of the palm tree (48 feet) and the horizontal distance from where you are lying on the beach to the base of the tree (60 feet). We need to find the angle of elevation from your position to the top of the tree.
Define the tangent function Let θ be the angle of elevation. We can use the tangent function to relate the angle of elevation to the height of the tree (opposite side) and the horizontal distance (adjacent side). The tangent function is defined as: tan ( θ ) = adjacent opposite
Calculate the angle of elevation In this case, the opposite side is the height of the tree (48 feet), and the adjacent side is the horizontal distance (60 feet). So we have: tan ( θ ) = 60 48 To find the angle of elevation θ , we need to take the inverse tangent (arctan) of 60 48 :
θ = arctan ( 60 48 ) θ = arctan ( 0.8 ) Using a calculator, we find that: θ ≈ 38.659 8 ∘
Round to the nearest tenth We need to round the angle to the nearest tenth of a degree. So, rounding 38.659 8 ∘ to the nearest tenth gives us 38. 7 ∘ .
Final Answer Therefore, the angle of elevation from your position to the top of the tree is approximately 38. 7 ∘ . The correct answer is d. 38. 7 ∘
Examples
Imagine you're building a ramp for skateboarding. You know the height you want to reach (the 'opposite' side) and the distance you have available to build the ramp horizontally (the 'adjacent' side). By calculating the arctangent of the ratio of height to distance, you can determine the angle of elevation needed for your ramp. This ensures the ramp isn't too steep or too gradual, making it safe and fun to use.
