Question

1. shandra has a greenhouse in her backyard. the pitch of the roof is 4:12. a) what is the slope of the roof? b) what is the angle of elevation? 6. one section of the crowsnest highway has a 6% grade. what will be the vertical change for a horizontal change of 15 km? 7. a) how long will a section of railroad track be if it rises 6.8 metres for a run of 125.9 metres? b) what is the percent grade of the road?

Explanation:

Step1: Define slope from pitch

The pitch of a roof is given as the ratio of vertical rise to horizontal run. For a pitch of 4:12, the slope $m$ is $\frac{4}{12}=\frac{1}{3}$.

Step2: Find angle of elevation

We know that slope $m = \tan\theta$, where $\theta$ is the angle of elevation. If $m=\frac{1}{3}$, then $\theta=\arctan(\frac{1}{3})$. Using a calculator, $\theta\approx 18.43^{\circ}$.

Step3: Solve for vertical change in highway

The grade is given as a percentage, and grade = $\frac{\text{vertical change}}{\text{horizontal change}}\times100$. If grade = 6% and horizontal change $x = 15$ km = 15000 m, then $0.06=\frac{y}{15000}$, and $y=0.06\times15000 = 900$ m.

Step4: Find length of railroad track

We can use the Pythagorean theorem for the railroad - track problem. The vertical rise $a = 6.8$ m and horizontal run $b = 125.9$ m. The length $L$ of the track is $L=\sqrt{a^{2}+b^{2}}=\sqrt{6.8^{2}+125.9^{2}}=\sqrt{46.24 + 15850.81}=\sqrt{15897.05}\approx126.08$ m.

Step5: Calculate percent grade of road

Percent grade = $\frac{\text{vertical change}}{\text{horizontal change}}\times100$. Here, vertical change $a = 6.8$ m and horizontal change $b = 125.9$ m. Percent grade=$\frac{6.8}{125.9}\times100\approx5.4\%$

Answer:

a) The slope of the roof is $\frac{1}{3}$.
b) The angle of elevation of the roof is approximately $18.43^{\circ}$.

1. The vertical change for a horizontal change of 15 km on the Crowsnest Highway is 900 m.
2. a) The length of the section of railroad track is approximately 126.08 m.

b) The percent grade of the road is approximately 5.4%.