Final answer:
To find the rise of the highway at a 6° incline over 8 miles, use the tangent of the angle which gives you the ratio of the rise to the run. Multiply the length of the run by the tangent of the angle to get the height of the rise. The result, 0.84 miles, is the height the highway rises over the 8-mile distance.
Step-by-step explanation:
The question you're asking involves using trigonometric functions to solve a problem related to angles of elevation. Given a highway that makes an angle of 6° with the horizontal over a distance of 8 miles, you wish to find how high the highway rises in this span.
To determine the height of the rise (h), we use the tangent function since it is the ratio of the opposite side to the adjacent side in a right triangle. Here the opposite side is the height of the rise and the adjacent side is the horizontal distance.
tangent(angle) = opposite/adjacent
Therefore:
tangent(6°) = h / 8 miles
Multiply both sides by 8 miles to isolate h:
h = 8 miles * tangent(6°)
h ≈ 8 miles * 0.1051
h ≈ 0.841 miles
Round h to the nearest hundredth:
h ≈ 0.84 miles