Traffic sign

There is a traffic sign for climbing on the road with an angle of 7%. Calculate at what angle the road rises (falls).

Correct result:

A =  4.004 °
B =  4.014 °

Solution:

$r=7 \%=\dfrac{ 7 }{ 100 }=0.07 \ \\ \ \\ \tan A=\dfrac{ y }{ x }=r \ \\ \ \\ A=\dfrac{ 180^\circ }{ \pi } \cdot \arctan(r)=\dfrac{ 180^\circ }{ \pi } \cdot \arctan(0.07)=4.004 ^\circ =4^\circ 15"$

$r=y/a \ \\ a=1 \ \text{m} \ \\ y_{1}=a \cdot \ r=1 \cdot \ 0.07=\dfrac{ 7 }{ 100 }=0.07 \ \text{m} \ \\ x_{1}=\sqrt{ a^2-y_{1}^2 }=\sqrt{ 1^2-0.07^2 } \doteq 0.9975 \ \text{m} \ \\ \ \\ B=\dfrac{ 180^\circ }{ \pi } \cdot \arctan(\dfrac{ y_{1} }{ x_{1} } )=\dfrac{ 180^\circ }{ \pi } \cdot \arctan(\dfrac{ 0.07 }{ 0.9975 } ) \doteq 4.014=4.014 ^\circ \doteq 4^\circ 50" \ \\ \ \\ B \approx A$

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