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irina1246 [14]
3 years ago
12

A road has a 10% grade, meaning increasing 1 unit of rise to every 10 units of run.

Mathematics
1 answer:
fenix001 [56]3 years ago
6 0

The grade is the ratio of rise to run, i.e. the slope aka the tangent.

\tan \theta = \dfrac{1}{10}

\theta = \arctan 0.1 \approx 5.711^\circ

Answer: (a) 6 degrees

For part b, the road is the hypotenuse c of a right triangle whose tangent of the small angle is 1/10.    The height h or rise is the side opposite the small angle.

\sin\theta = \dfrac h c

h = c \sin \theta

We could just take the sine of the angle we got but let's get it from the tangent exactly.

\cos^2 \theta + \sin ^2 \theta = 1

Dividing by squared cosine

1 + \tan ^2 \theta = 1/\cos^2 \theta = 1/(1- \sin^2 \theta)

(1- \sin^2 \theta) = 1/(1 + \tan^2 \theta)

\sin^2 \theta = 1 - 1/(1 + \tan^2 \theta)

\sin^2 \theta = 1 - 1/(1 + (1/10)^2) = 1-1/(101/100) = 1/101

h = c \sin \theta = 2 \sqrt{1/101} \approx 0.199

Answer: (b) Rise of 0.199 km

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