Final Answer:
It corresponds to the calculated height of the tree using the provided information and the law of sines. Therefore the correct option is C) 232.1 ft
Step-by-step explanation:
To determine the height of the tree, we can use the law of sines by setting up a triangle with the given information. In this scenario, we have a right triangle formed by the tree, the road, and the line of sight from the observer.
The observer is 500 feet downhill, forming the base of the triangle. The angle of inclination from the road to the top of the tree is 5.5 degrees, and the road has a 7.1% grade, implying a 7.1-degree angle of inclination with the horizontal.
Using the law of sines: sin(5.5°) / height of tree = sin(7.1°) / hypotenuse (distance from observer to the top of the tree).
Rearranging the formula to solve for the height of the tree gives: height of tree = (sin(5.5°) * hypotenuse) / sin(7.1°).
First, find the hypotenuse using trigonometry: hypotenuse = distance from observer / cos(7.1°).
hypotenuse = 500 / cos(7.1°) ≈ 506.4 feet.
Now, plug this into the law of sines formula: height of tree = (sin(5.5°) * 506.4) / sin(7.1°) ≈ 232.1 feet.
Therefore, the approximate height of the tree is 232.1 feet to the nearest tenth of a foot.
Therefore the correct option is C) 232.1 ft