Question

practice quiz 4 - grade and pitch

1. one of the steepest railways in the world is the lisbon tram in portugal. in one section, it has a grade of 20%.

a) express this as a slope in fraction form
b) calculate the angle of elevation
c) what is the rise for a run of 10 m? (set up and solve a proportion)

1. the pitch of a roof on a lean - to that dianna is building is 2:5. if the lean - to touches the ground 4.8m from the base of the building, how high up the building does it reach? (set up and solve a proportion)

Explanation:

Step1: Convert grade to fraction

The grade is given as 20%. Grade is the ratio of rise to run expressed as a percentage. To convert 20% to a fraction, we write it as $\frac{20}{100}=\frac{1}{5}$.

Step2: Calculate angle of elevation

We know that slope $m = \tan\theta$, where $\theta$ is the angle of elevation. Since $m=\frac{1}{5}$, then $\theta=\arctan(\frac{1}{5})$. Using a calculator, $\theta=\arctan(0.2)\approx 11.31^{\circ}$.

Step3: Set up and solve proportion for rise

Let the rise be $y$ and run $x = 10$m. Since the slope $m=\frac{1}{5}$, and $m=\frac{y}{x}$, we have the proportion $\frac{y}{10}=\frac{1}{5}$. Cross - multiplying gives $5y = 10$, so $y = 2$m.

Step4: Solve for height of lean - to

The pitch of the roof is 2:5, which means the slope $m=\frac{2}{5}$. Let the height up the building be $h$ and the horizontal distance $x = 4.8$m. Since $m=\frac{h}{x}$, we set up the proportion $\frac{h}{4.8}=\frac{2}{5}$. Cross - multiplying gives $5h=2\times4.8 = 9.6$. Then $h=\frac{9.6}{5}=1.92$m.

Answer:

1.
a. $\frac{1}{5}$
b. Approximately $11.31^{\circ}$
c. 2m

1. 1.92m