Q$4(25$ points$)$

A sag curve is being constructed in a rural setting to transition between a horizontal section $(G=0\%)$ and a section with a $6\%$ grade. The posted speed limit is $80k\frac{m}{h}r.$ The initial design does not include streetlights on the curve. The coefficient of friction of the road is $0\mathrm{.}32$ and the headlights are $0\mathrm{.}5$ m off of the ground.

i$)$ Using the following equation for headlight control, what length of curve is required?

ii$)$ Using the K$-$values for headlight control $($Table $3\mathrm{.}5$ from Vertical Curves handout$)$ what is the required length? What might cause the difference with the previous value? iii$)$ If street lights were built on the section, what would the required length be$?$