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 percentage to slope

The grade percentage is the ratio of rise to run as a percentage. To convert 13.5% to a slope (m), we write it as a fraction. Since 13.5%=$\frac{13.5}{100}$, and slope $m = \frac{\text{rise}}{\text{run}}$, so $m=0.135$.

Step2: Calculate angle of elevation

We know that $\tan\theta=m$, where $\theta$ is the angle of elevation and $m$ is the slope. Given $m = 0.135$, then $\theta=\arctan(0.135)$. Using a calculator, $\theta=\arctan(0.135)\approx7.69^{\circ}$.

Step3: Find rise for a given run

We know that $m=\frac{\text{rise}}{\text{run}}$. Given $m = 0.135$ and $\text{run}=15$ m. Rearranging the formula for rise gives $\text{rise}=m\times\text{run}$. So $\text{rise}=0.135\times15 = 2.025$ m.

Answer:

a) 0.135
b) Approximately $7.69^{\circ}$
c) 2.025 m