Final answer:
To find the height of the tree, we use the angle of elevation and the length of the zip line with trigonometric functions, specifically the tangent and cosine, to calculate the height based on the angle and horizontal distance, assuming the zip line ends at ground level.
Step-by-step explanation:
To determine how tall the tree is, given an angle of elevation to the top of the tree from the ground is 51° and the zip line from the tree to the campsite is 145 feet long, we can use trigonometry. Specifically, we'll use the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, the opposite side is the height of the tree, and the adjacent side is the horizontal distance from the base of the tree to the point on the ground directly below the end of the zip line.
The tangent of the angle of elevation (51°) is equal to the height of the tree divided by the horizontal distance. However, we are not given the horizontal distance directly, and since we do not have a right-angled triangle, we cannot directly apply the tangent function. Instead, we need to make an assumption or infer additional information. If we assume the end of the zip line is at ground level, then the horizontal distance would be the hypotenuse (the length of the zip line) multiplied by the cosine of the angle of elevation. This can be solved with the equation: height = (tangent of 51°) × (145 feet × cosine of 51°).