The gradient of a road: percent and angle

The plane and space, grade 11 (3 units) Lesson 12 of 15Learning

The gradient of a road: percent and angle

A gradient in percent

A sign on a mountain road says 8%: for every 100 m measured horizontally, the road climbs (or drops) 8 m.

The gradient in percent is the climb over the horizontal distance, times 100.
The climb is opposite the angle $\displaystyle \alpha$ between the road and the horizontal, and the horizontal distance is next to it – so their ratio is $\displaystyle \tan\alpha$ .

The gradient in percent, $\displaystyle p$ :

\displaystyle p

\displaystyle =

\displaystyle 100\cdot\tan\alpha

A road of 8%: $\displaystyle {\tan\alpha}={0.08}$ , so $\displaystyle {\alpha}={\tan^{-1}0.08\approx4.57^\circ}$ .
A gradient of 100% is an angle of $\displaystyle 45^\circ$ , not $\displaystyle 90^\circ$ : a climb of 100 m over 100 m horizontally.

Exercise 1

What is the angle between the road and the horizontal? Round to two decimal places.

A gradient of 12%

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A gradient of 10%

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A stretch of road climbs 45 m over 600 m measured horizontally.

What is its gradient in percent?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the angle between the road and the horizontal? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

An access ramp

A ramp is comfortable for a wheelchair when its gradient is at most $\displaystyle 1:12$ : a climb of one metre for every 12 m horizontally.

$\displaystyle \frac{1}{12}\approx0.083$ , a gradient of about 8.3%.
The angle: $\displaystyle \tan^{-1}\frac{1}{12}\approx4.76^\circ$ .

Exercise 3

A building’s entrance is 0.6 m above the pavement.
A ramp with a gradient of $\displaystyle 1:12$ is built up to it.

What is the ramp’s horizontal length, in metres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How long is the ramp itself, in metres? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A car drives 2 km along a road with a gradient of 6%.
The distance is measured along the road, by the car’s odometer.
How many metres did the car climb? Find the angle first, and round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Which road is steeper: a road with a gradient of 20%, or a road at $\displaystyle 15^\circ$ to the horizontal?
Explain.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 11 (3 units)

Open as a page

Scale

The real length – $\displaystyle l$ the length on the map, at a scale of $\displaystyle 1:n$
$\displaystyle L$ $\displaystyle =$ $\displaystyle n\cdot l$
The real area – $\displaystyle s$ the area on the map, at a scale of $\displaystyle 1:n$
$\displaystyle S$ $\displaystyle =$ $\displaystyle n^{2}\cdot s$

Similar shapes

The scale factor – $\displaystyle a_1$ , $\displaystyle a_2$ corresponding sides
$\displaystyle k$ $\displaystyle =$ $\displaystyle \frac{a_2}{a_1}$
The ratio of the areas of similar shapes
$\displaystyle \frac{S_2}{S_1}$ $\displaystyle =$ $\displaystyle k^{2}$
A height by its shadow – $\displaystyle H$ , $\displaystyle S$ the object and its shadow; $\displaystyle h$ , $\displaystyle s$ a stick and its shadow
$\displaystyle \frac{H}{S}$ $\displaystyle =$ $\displaystyle \frac{h}{s}$

Trigonometry in the right-angled triangle

Sine – $\displaystyle a$ the side opposite the angle, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
Cosine – $\displaystyle b$ the side adjacent to the angle
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
Tangent – the opposite side over the adjacent side
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle from its tangent (and likewise $\displaystyle \sin^{-1}$ , $\displaystyle \cos^{-1}$ )
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \tan^{-1}\frac{a}{b}$
Pythagoras’ theorem
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
A gradient in percent – $\displaystyle \alpha$ the angle with the horizontal
$\displaystyle p$ $\displaystyle =$ $\displaystyle 100\cdot\tan\alpha$