Larissa dives into a pool that is 8 feet deep. She touches the bottom of the pool with her hands 6 feet horizontally from the point at which she entered the water.
What is the approximate angle of elevation from the point on the bottom of the pool where she touched to her entry point?
A. $36.9^{\circ}$
B. $41.4^{\circ}$
C. $48.6^{\circ}$
D. $53.1^{\circ}$
Answer (2)
Define the angle of elevation as θ , recognize the depth of the pool as the opposite side (8 feet) and the horizontal distance as the adjacent side (6 feet).
Express the relationship using the tangent function: tan ( θ ) = 6 8 = 3 4 .
Calculate the angle θ by taking the inverse tangent (arctan) of 3 4 .
Approximate the angle of elevation: 53. 1 ∘ .
Explanation
Analyze the problem Let's analyze the problem. Larissa dives into a pool and touches the bottom at a certain horizontal distance from her entry point. We are given the depth of the pool (8 feet) and the horizontal distance (6 feet). We need to find the angle of elevation from the point where she touched the bottom to her entry point.
Set up the tangent equation Let θ be the angle of elevation we want to find. The depth of the pool is the opposite side of the angle θ , which is 8 feet. The horizontal distance is the adjacent side of the angle θ , which is 6 feet. We can use the tangent function to relate the angle to the opposite and adjacent sides: tan ( θ ) = adjacent opposite = 6 8 = 3 4
Calculate the angle To find the angle θ , we need to take the inverse tangent (arctan) of 3 4 :
θ = arctan ( 3 4 ) Using a calculator, we find that: θ ≈ 53.1 3 ∘ Rounding to one decimal place, we get θ ≈ 53. 1 ∘ .
State the final answer Therefore, the approximate angle of elevation from the point on the bottom of the pool where she touched to her entry point is 53. 1 ∘ .
Examples
Understanding angles of elevation is crucial in many real-world applications, such as surveying, navigation, and construction. For instance, when building a ramp, knowing the angle of elevation helps ensure it meets safety standards and is accessible. Similarly, in navigation, pilots use angles of elevation to determine their approach path to an airport. These examples highlight how trigonometry and angles of elevation play a vital role in ensuring safety and precision in various fields.
The approximate angle of elevation from the bottom of the pool to the entry point is 53.1 degrees. This is calculated using the tangent function based on the vertical and horizontal distances involved. Therefore, the correct answer is D. 53.1°.
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