Question

1. one of the steepest railways in the world is the lisbon tram in portugal. in one section, it has a grade of 13.5%.

a) express this as a slope.
b) calculate the angle of elevation.
c) what is the rise for a run of 15 m?

Explanation:

Step1: Convert grade to slope

Grade is defined as $\frac{\text{rise}}{\text{run}}\times100\%$. Given grade = $13.5\%=\frac{13.5}{100}$. So the slope $m = 0.135$.

Step2: Calculate angle of elevation

We know that slope $m=\tan\theta$, where $\theta$ is the angle of elevation. So $\theta=\arctan(m)$. Substituting $m = 0.135$, we get $\theta=\arctan(0.135)\approx7.7^{\circ}$.

Step3: Find rise for a given run

Since slope $m=\frac{\text{rise}}{\text{run}}$, and $m = 0.135$, run $=15$ m. Then rise $=m\times\text{run}=0.135\times15 = 2.025$ m.

Answer:

a) The slope is $0.135$.
b) The angle of elevation is approximately $7.7^{\circ}$.
c) The rise for a run of $15$ m is $2.025$ m.