The state highway commission has decided to build a bridge over a dangerous railroad crossing. The approach ramp has to be constructed so that the bridge is 10 meters above the track. If the maximum desired grade of this ramp is 4 degrees, how far back from the edge of the bridge must the ramp start?
look at it as a triangle
Angle is 4 degree
tan of 4 =
Side opposite(10m)/ Side adjacent
distance required = Side adjacent =
10 / tan 4
To determine how far back from the edge of the bridge the ramp must start, we need to consider the height difference and the maximum desired grade.
Let's break down the problem into steps:
Step 1: Convert the maximum desired grade from degrees to a decimal.
The maximum desired grade is given as 4 degrees. To convert it to a decimal, we divide it by 100:
4 degrees / 100 = 0.04 (as a decimal)
Step 2: Determine the horizontal distance required to achieve the desired elevation change.
Since the bridge needs to be 10 meters above the track and the maximum desired grade is 0.04, we can use trigonometry to find the horizontal distance (x) required.
We can use the tangent function to determine the relationship between the height difference and the horizontal distance:
tan(angle) = opposite/adjacent
tan(angle) = 10 meters / x
Substituting the values:
tan(0.04) = 10 meters / x
Step 3: Solve the equation to find the horizontal distance.
To solve for x, we rearrange the equation:
x = 10 meters / tan(0.04)
Using a calculator, evaluate tan(0.04) and divide it into 10 meters to find the value of x.
Calculating tan(0.04):
tan(0.04) ≈ 0.0006981
Now, we can find x:
x = 10 meters / 0.0006981
Using a calculator, divide 10 meters by 0.0006981 to find the value of x.
x ≈ 14,321 meters
Therefore, the ramp must start approximately 14,321 meters back from the edge of the bridge.