Question

1. determine the angle of elevation of a road with each grade. express your answers to the nearest degree. a) 16% b) 2% c) 6% d) 20%

Explanation:

Step1: Recall grade - angle relationship

The grade of a road is the ratio of the vertical rise to the horizontal run, expressed as a percentage. If the angle of elevation is $\theta$, then the grade $g$ is given by $g = 100\times\tan\theta$. So, $\tan\theta=\frac{g}{100}$.

Step2: Calculate angle for part a

For $g = 16\%$, we have $\tan\theta=\frac{16}{100}=0.16$. Then $\theta=\arctan(0.16)$. Using a calculator, $\theta\approx9^{\circ}$.

Step3: Calculate angle for part b

For $g = 2\%$, we have $\tan\theta=\frac{2}{100}=0.02$. Then $\theta=\arctan(0.02)$. Using a calculator, $\theta\approx1^{\circ}$.

Step4: Calculate angle for part c

For $g = 6\%$, we have $\tan\theta=\frac{6}{100}=0.06$. Then $\theta=\arctan(0.06)$. Using a calculator, $\theta\approx3^{\circ}$.

Step5: Calculate angle for part d

For $g = 20\%$, we have $\tan\theta=\frac{20}{100}=0.2$. Then $\theta=\arctan(0.2)$. Using a calculator, $\theta\approx11^{\circ}$.

Answer:

a) $9^{\circ}$
b) $1^{\circ}$
c) $3^{\circ}$
d) $11^{\circ}$