The grade of a road or a railway road bed is the ratio rise/run, usually expressed as a percent. For example, a railway with a grade of 5% rises 5 ft for every 100 ft of horizontal distance.
1. The Johnstown, Pennsylvania, inclined railway was built as a "lifesaver" after the Johnstown flood of 1889. It has a 987-ft run at a 71% grade. How high does this railway lift its passengers?
2. The Fenelon Place Elevator railway in Dubuque, Iowa, lifts passengers 189 ft to the top of a bluff. It has an 83% grade. How long is this railway?
3. The Duquesne Incline Plane Company's roadway in Pittsburgh, Pennsylvania, climbs Mt. Washington, located above the mouth of the Monongahela River. It reaches a height of 400 ft with a 793-ft incline. What is its grade?
200%
To solve each of these problems, we will use the formula for grade:
Grade = (Rise/Run) * 100
1. For the Johnstown inclined railway:
Given: Run = 987 ft, Grade = 71%
To find: Rise
We can rearrange the formula to solve for the rise:
Rise = (Grade/100) * Run
Substituting the given values:
Rise = (71/100) * 987
Rise = 700.77 ft
Therefore, the railway lifts its passengers to a height of approximately 700.77 ft.
2. For the Fenelon Place Elevator railway:
Given: Grade = 83%, Rise = 189 ft
To find: Run
Again, rearranging the formula to solve for run:
Run = Rise / (Grade/100)
Substituting the given values:
Run = 189 / (83/100)
Run = 227.71 ft
Therefore, the length of the Fenelon Place Elevator railway is approximately 227.71 ft.
3. For the Duquesne Incline Plane Company's roadway:
Given: Rise = 400 ft, Run = 793 ft
To find: Grade
Using the given values in the formula:
Grade = (400/793) * 100
Grade = 50.44%
Therefore, the grade of the Duquesne Incline Plane Company's roadway is approximately 50.44%.