Per Cent Grade - Railroad Track

[Guest]

A railroad track's grade is specified in per cent. A grade of 1% has a rise of 1 ft. in 100 ft. The total rise in a 1 3/4 % grade is x. Calculate the total amount of track with this grade.

I have the answer to both questions - I don't understand how to calculate them. Thanks.

[Guest]

Please reply - Thanks.

[Guest]

A grade of 1 3/4 % is 1 3/4 ft in 100 feet OR a rise of 1 foot in 57.142 feet.

So to rise "X" then run of track is 57.142 x "X" = 57.142X feet measured horizontal

The sloping track will be sqroot [ X^2 + (57.142^2 x X^2) ] = 57.15X feet

[Guest]

According to answer provided with problem:

Total rise in a 1 3/4 % grade - 43.6 ft.

Length of track with grade - 2491 3/7 ft.

[Guest]

If the total rise "X" in a 1 3/4 % grade - 43.6 ft.
then 57.142 x "X" = 57.142 x 43.6 = 2491.39 ft
Length of track with grade - 2491 3/7 ft.

[Guest]

"A grade of 1 3/4 % is 1 3/4 ft in 100 feet OR a rise of 1 foot in 57.142 feet." How was the 43.6 ft. calculated ?

"The sloping track will be sqroot [ X^2 + (57.142^2 x X^2) ] = 57.15X feet" I know what you are saying - I don't understand the calculations.

Thanks.

[Guest]

From the original question........."The total rise in a 1 3/4 % grade is x."

For a particular problem we need to know a value for "X"..........

Then you gave this in your posting......"Total rise in a 1 3/4 % grade - 43.6 ft".......which I took for the value of "X" and worked out the corresponding length of track.

The grade of 1 3/4 % is simply a slope.... the further you go up the slope the more you will rise. So need to know where to calculate to.

The rise is a vertical measurement. The grade is calculated as ...."rise" divided by "run" OR rise / run.
The "run" is the corresponding horizontal measurement for a particular rise. eg 1 % is 1 foot rise per 100 foot run.

The track is laid along the rising slope.......consider it all as a right angled triangle and use Pathagoras to calculate the length of the sloping track.
but because the angle is very small the sloping side is almost the same length as the horizontal and for approximation can be taken as equal.
That makes the Tan of the angle approx equal to the Sin of the angle.

[Guest]

Thanks.

I know Pathagoras has the theorem a^2 + b^2 = c^2 - "sqroot [ X^2 + (57.142^2 x X^2) ] = 57.15X feet" Would that be the theorem ?

I have heard of Tan and Sin but I didn't take a Trig course.

[Guest]

I know Pathagoras has the theorem a^2 + b^2 = c^2 - "sqroot [ X^2 + (57.142^2 x X^2) ] = 57.15X feet" Would that be the theorem ?

Let "a" equal the "rise" X .... then a^2 = X^2

Let "b" equal the "run" in this case 57.142X ......so b^2 = (57.142X)^2 = (57.142^2) x (X^2).

Let c = the sloping side of the right angle triangle ....the actual track itself...
Let a^2 + b^2 = c^2 ....then... [X^2 + (57.142^2 x X^2)] = [(57.142^2 + 1) x (X^2)]

and then "c" = sqroot [X^2 + (57.142^2 x X^2)] = sqroot [(57.142^2 + 1) x (X^2)] = 57.151X feet

In this particular question if rise "X" is 43.6 feet then the sloping side is 57.151 x 43.6 = 2491.78 feet.

[Guest]

Thanks.

[Guest]

I have some questions:

The 1% grade has a rise of 1 ft. in 100 ft. The 1 3/4 % has a total rise of 43.6 ft.

1) Could the total rise be calculated by proportion ?

2 ) Is the 1 3/4 % indicating a rise of 1.75 ft. in 100 ft. ?

Thanks again.

[Guest]

Yes, a 1% grade is 1 foot in 100 feet so will rise 43.6 feet in (43.6 / 1) x 100 = 4360 feet measured horizonal as the "run".
By ratio or proportion corresponding sides ratios equal so .....43.6 / 1 = "run" /100 ....so "run" = 43.6 x 100 = 4360 feet horizontal

and 1 3/4% grade is 1.75 feet in 100 feet so ..... 43.6 / 1.75 = "run" / 100 ....which gives.... 43.6 x 100 = 1.75 x "run"
then....."run" = 43.6 x 100 /1.75 wich gives .... 2491.428 OR 2491.43 feet to 2dp.


.... and this calculation does not have the rounding errors that were introduced in the earlier calculation when we worked out the "run" for a 1 foot rise and then multiplied by 43.6.

[Guest]

Thanks again.

[Guest]

So, for the "rise" and "run":

A 1% grade in 100 feet is 0.01 ft. rise in 1 ft. run.

A 1 3/4% grade in 100 feet is 0.0175 ft. rise in 1 ft. run.

Correct ?

[Guest]

Please respond - Thanks.

[Guest]

yes, you have just divided the dimensions by 100. the angle of the gradient is the same.
.....as in similar triangles

[Guest]

Thanks.