Mountain Descent
A road sign at the top of a mountain indicates that for the next 4 miles the grade is 12%. Find the angle of the grade and the change in elevation for a car descending the mountain.
arctan .12 = 6.8°
the car ascends 12 ft for every 100 ft of horizontal travel, so it rises
.12 * 4 * 5280 = 2534 ft
To find the angle of the grade, we can use the inverse tangent function. The inverse tangent, often written as atan or tan^(-1), gives us the angle when we know the ratio of the opposite and adjacent sides of a right triangle. In this case, the opposite side would be the change in elevation and the adjacent side would be the horizontal distance.
First, let's find the change in elevation. The road sign tells us that the grade is 12% for the next 4 miles. To convert miles to feet, we need to multiply by 5280 (since 1 mile = 5280 feet):
Change in Elevation = 4 miles * 5280 feet/mile = 21120 feet
Now, we can use trigonometry to find the angle. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle. In this case, the tangent of the angle will be equal to the change in elevation divided by the horizontal distance.
Tangent(angle) = Change in Elevation / Horizontal Distance
Let's find the horizontal distance first. To do this, we can use the formula for finding the hypotenuse of a right triangle, which states that the hypotenuse is equal to the square root of the sum of the squares of the other two sides.
Horizontal Distance = sqrt((Change in Elevation / Grade)^2 + (Change in Elevation)^2)
Using the values we have, we can calculate the horizontal distance:
Horizontal Distance = sqrt((21120 / 0.12)^2 + 21120^2) = sqrt((176000)^2 + (21120)^2) = sqrt(309760000 + 446054400) = sqrt(755814400) ≈ 27492 feet
Now, we can find the angle:
Tangent(angle) = Change in Elevation / Horizontal Distance
Tangent(angle) = 21120 / 27492
To find the angle, we take the inverse tangent of both sides:
Angle = atan(21120 / 27492)
Using a calculator or trigonometric table, we can find the angle to be approximately 35.5 degrees.
Therefore, the angle of the grade is approximately 35.5 degrees and the change in elevation for a car descending the mountain is 21120 feet.