Question

In geometry Roberts' engineering team is developing a plan for constructing a new straight road to the top of a hill in his town. He needs to ensure that the grade or the angle of elevation the road forms with the flat road is 6°. Given that the height of the hill is 100 feet, what is the approximate length of the road in feet?

Answer 1

The approximate length of the road needed to maintain a 6° angle of elevation to reach the top of a 100-foot hill is about 951.49 feet, calculated using the tangent function.

Step-by-step explanation:

To calculate the approximate length of the road to the top of a hill when the angle of elevation is 6° and the height of the hill is 100 feet, we use trigonometric functions.

Specifically, we use the tangent function, which relates the angle of a right triangle to the ratios of its opposite side over its adjacent side.

Since the height of the hill is the opposite side and the length of the road is the hypotenuse in this scenario, we can set up the equation tan(6°) = 100 feet / length of the road.

From this, rearranging the equation to solve for the length of the road, we find:

length of the road = 100 feet / tan(6°).

Using a calculator, tan(6°) is approximately 0.1051. Hence:

length of the road ≈ 100 / 0.1051 ≈ 951.49 feet.

Therefore, the approximate length of the road needed to maintain a 6° angle of elevation to the top of a 100-foot hill is 951.49 feet.