Solving a right triangle (Trigonometry)

Question

A highway has a gradient of 12.8% (i.e. it rises 12.8 m in 100 m horizontally). How much would a car rise (in m) if it drove 31 m along the highway?

Round answer to 2 decimal places.

Answer 1

Defining Variables

1. Let x represent how far the car moves in the horizontal direction
2. Let y represent how far the car moves in the vertical direction
3. Let h represent the hypotenuse of the right triangle formed by x and y
4. Let m represent the gradient of the highway

Goal

find y when:

1. h = 31
2. m = 12.8 / 100 = 0.128

Applicable equations

1. Gradient = rise / run = y / x

2. x² + y² = h² (Pythagorean theorem)

Plugging in known data into applicable equations

1. 0.128 = y / x or x = y / 0.128

2. x² + y² = 31²

Plugging in equation 1 into equation 2 and solving for y

*any negative solutions are disregarded

$$(y / 0.128)^2 + y^2 = 31^2$$ $$(y^2 / 0.016384) + y^2 = 31^2$$ $$61.03515625y^2 + y^2 = 31^2$$ $$62.03515625y^2 = 31^2$$ $$y^2 = 31^2/62.03515625$$ $$y^2 = 15.4912159184$$ $$y = 3.93588819943$$ $$y ≈ 3.94m$$