A road has a 10% grade (slope). To the nearest degree what is the angle of elevation of the road?
should I use sin?
rise = .1 run
tan theta = .1
theta = 5 deg
thank you, I was confused
Yes, you can use the sine function to find the angle of elevation of the road. The angle of elevation is the angle between the horizontal ground and the inclined road. The opposite side of the triangle formed by the road and the ground is the vertical rise, and the adjacent side is the horizontal distance.
To find the angle of elevation using the sine function, you can use the following formula:
sin(angle) = opposite/hypotenuse
In this case, the opposite side is the vertical rise (the rise in elevation on the road), and the hypotenuse is the distance along the road. The adjacent side will be the horizontal distance. Since the grade is given as a percentage, you can use it to calculate the vertical rise and the horizontal distance.
To find the angle of elevation, follow these steps:
Step 1: Convert the grade percentage to a decimal. For example, a 10% grade is equivalent to 0.10.
Step 2: Calculate the vertical rise. Multiply the decimal grade by the horizontal distance. For example, if the horizontal distance is 100 meters, then the vertical rise would be 0.10 * 100 = 10 meters.
Step 3: Calculate the hypotenuse (distance along the road) using the Pythagorean theorem. The hypotenuse is the square root of the sum of the squares of the vertical rise and the horizontal distance. In our example, if the horizontal distance is 100 meters and the vertical rise is 10 meters, then the hypotenuse would be sqrt(100^2 + 10^2) = sqrt(10000 + 100) = sqrt(10100) ≈ 100.49875 meters.
Step 4: Plug the values of the vertical rise and the hypotenuse into the sine function:
sin(angle) = opposite/hypotenuse
sin(angle) = 10/100.49875
Step 5: Solve for the angle by taking the inverse sine (sin^-1) of the ratio:
angle = sin^-1(10/100.49875)
Step 6: Calculate the angle using a calculator or a trigonometric table. The result will be the angle of elevation in radians.
Finally, convert the angle from radians to degrees by multiplying by 180/π (pi) to get the angle of elevation in degrees.