Problem

A ramp rises \(3~\text{ft}\) over a horizontal run of \(12~\text{ft}\). Form \(\tan(θ)\) as rise over run, simplify the ratio, and interpret it as feet of rise per horizontal foot.

Prompt diagram of a ramp forming a right triangle, with reference angle theta at ground level, vertical rise 3 feet, horizontal run 12 feet, and the ramp as the unlabeled hypotenuse. The trig ratio is withheld.

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What this problem is really about

At the ground angle, a ramp’s rise and run are the opposite and adjacent legs. We’ll form tangent as rise over run, simplify while noting that the units cancel in the ratio, and then translate that dimensionless value back into a clear feet-of-rise per horizontal-foot rate.

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sin(θ) = 3/√153, so the ramp rises 3 feet per ramp length.

tan(θ) = 3/12 = 1/4, so the ramp rises 1 foot per 4 horizontal feet.

tan(θ) = 1/4 foot.

tan(θ) = 12/3 = 4, so the ramp rises 4 feet per horizontal foot.

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A ramp rises \(3~\text{ft}\) over a horizontal run of \(12~\text{ft}\). Form \(\tan(θ)\) as rise over run, simplify the ratio, and interpret it as feet of rise per horizontal foot.

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