Grade 10 Math · Grades and slopes
One hill, three costumes
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One hill, three costumes

Chapter 2 called it slope. Road signs call it grade: G=100ΔhLG = \dfrac{100\,\Delta h}{L}G equals one hundred times the rise, over the run, where Δh\Delta h (delta-h) is the change in height, LL is the LEVEL run in the same unit, and GG is the grade itself, in percent. An 8% hill climbs 8 m for every 100 m of LEVEL run. Earthworks crews wear a third costume: an n:1 slope, n metres across for each 1 down, so G=100nG = \dfrac{100}{n} — a gentle 4:1 bank is a 25% grade.

And here is this chapter’s handshake: rise over run is exactly tanθ\tan\theta. So G=100tanθG = 100\tan\theta, and θ=tan1 ⁣(G100)\theta = \tan^{-1}\!\left(\dfrac{G}{100}\right) turns any grade into its angle. A 100% grade is not a wall — it is tan1(1)=45\tan^{-1}(1) = 45^\circ. Say that at dinner and watch somebody argue. You will win.