Surveyors with a highway planning crew measure a section of proposed roadway to be 4.4 km. They measure the angle of inclination over the 4.4 km to be 8.5 degrees. Determine the change in altitude for drivers travelling from one end to the other. Answer to the nearest whole kilometre.
To determine the change in altitude for drivers traveling from one end of the proposed roadway to the other, we can use trigonometry. Specifically, we can use the trigonometric function tangent (tan).
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. In this case, the angle of inclination (8.5 degrees) represents the angle between the horizontal surface (adjacent side) and the roadway (opposite side).
Let's denote the change in altitude as "h" (in km). We can use the formula:
tan(angle) = opposite/adjacent
In this case, the opposite side is the change in altitude (h), and the adjacent side is the length of the roadway (4.4 km). Substituting these values:
tan(8.5 degrees) = h/4.4 km
To solve for "h," we can rearrange the equation:
h = tan(8.5 degrees) * 4.4 km
Now, we can calculate the change in altitude using this formula:
h ≈ tan(8.5 degrees) * 4.4 km
Using a scientific calculator or an online calculator, type in "tan(8.5)" and multiply the result by 4.4:
h ≈ 0.1504 * 4.4 km
h ≈ 0.6626 km
Therefore, the change in altitude for drivers traveling from one end to the other is approximately 0.6626 km.