Question

1. jon is building a wheelchair ramp at his grandmothers house. the ramp will rise 1.2 m over a run of 7.2 m. jon will purchase roll roofing to lay on the ramp to create a non - slip surface.

a) what length of roll roofing will jon have to purchase to cover the wheelchair ramp?
b) what is the angle of elevation of the ramp?

1. akiko works as a ski instructor. he has read that human - triggered avalanches occur most often on slopes with angles of elevation between 35° and 45°.

a) what are the slopes of these two angles of elevation?
b) what are the percent grades? whether an avalanche will occur depends on many factors, including the slope of the hill, the moisture content in the snow, and the temperature.

Explanation:

Step1: Solve 2a - Use Pythagorean theorem

The ramp forms a right - triangle with height (rise) $a = 1.2$ m and base (run) $b = 7.2$ m. The length of the ramp (hypotenuse $c$) is given by $c=\sqrt{a^{2}+b^{2}}=\sqrt{1.2^{2}+7.2^{2}}=\sqrt{1.44 + 51.84}=\sqrt{53.28}\approx7.3$ m.

Step2: Solve 2b - Use tangent function

Let the angle of elevation be $\theta$. $\tan\theta=\frac{\text{rise}}{\text{run}}=\frac{1.2}{7.2}=\frac{1}{6}$. Then $\theta=\arctan(\frac{1}{6})\approx9.5^{\circ}$.

Step3: Solve 3a - Recall slope formula

The slope $m$ of a line with angle of elevation $\alpha$ is $m = \tan\alpha$. For $\alpha = 35^{\circ}$, $m=\tan35^{\circ}\approx0.70$. For $\alpha = 45^{\circ}$, $m=\tan45^{\circ}=1$.

Step4: Solve 3b - Convert slope to percent - grade

Percent - grade is the slope expressed as a percentage. For a slope $m$, percent - grade $=m\times100\%$. For $m = \tan35^{\circ}\approx0.70$, percent - grade $\approx0.70\times100\% = 70\%$. For $m=\tan45^{\circ}=1$, percent - grade $=1\times100\% = 100\%$.

Answer:

a) Approximately 7.3 m
b) Approximately 9.5°

1. a) For 35°, slope is approximately 0.70; for 45°, slope is 1

b) For 35°, percent - grade is approximately 70%; for 45°, percent - grade is 100%