# Inclined lines can have the same slope

## Meinstein

### Incline on roads

The greatest gradient of a PostBus route in Europe can be found in the **Kiental**, Canton of Bern, Switzerland. It is given as 28%. What does that mean?

To better understand slopes, let's draw a slope triangle.

### Definition of the slope with the slope triangle

The **Slope m** is defined as the quotient of height difference h and length difference l. In the Kiental, this is 28m height difference per 100m at the steepest points.

Become**Gradients in percent** specified, that means the number of meters in altitude per 100 meters. Then the gradient can be read off directly in percent. A slope m from any values of h and l must therefore still be multiplied by 100 in order to get the slope m in percent.

**Notice**: a **100% gradient** is not vertical! According to the definition, it is a height difference of 100m to 100m horizontal difference.

The vertical “slope” has the value m = infinite.

Gradients are negative gradients, their m is negative.

### Values and signs for slopes

### Tasks with solutions

The difference in altitude is 180m. The road measures 4.5km on the map. How big is the slope in percent?

Solution: slope = 4%

Places A and B have a height difference of 250m. A maximum gradient of 12% is recommended for roads. So how long will the road between these places be at least?

Solution: Length of the street at least: 2083.33m

The Jungfrau Railway has a length of 9.34km. From the Kleine Scheidegg to the Jungfraujoch at 3,454 meters above sea level, it has an average slope of 14.9143%. How high is the little Scheidegg?

Solution: Kleine Scheidegg: 2061 m above sea level.

The Öschibach flows 2.18 km from 1410 m above sea level to 1176 m above sea level. How big is its slope in percent?

Solution: slope = 10.73%

Enter the gradient in% between these points of the Swiss map coordinate network:

(616600/166100/400)

(614200/166100/1380)

Solution: slope = 40.83%

### Slope of straight lines in the coordinate system

The slope factor plays an important role in the linear functions:

In the coordinate system, slopes (of straight lines, i.e. linear functions) are defined in exactly the same way:

Slope m

m = height difference / horizontal distance

m = Δy: Δx

here m = 3: 4 = 0.75

m has no units (the distance units are shortened.

Generally in the coordinate system:

Two points define exactly one straight line (linear function), which is the slope **m** Has.

### Exercises and solutions for slope in the coordinate system

How big is the slope between points P1 (2/2) and P2 (0/4)?

Solution: m = -1

### Solving slopes with the tangent (angle function)

If slopes are specified with angles, the remaining parts of a slope triangle can be solved with the help of trigonometric functions:

Correspondingly, the adjacent cathete can be calculated with the cotangent and the hypotenuse with the sine.

### Tasks and solutions of incline tasks with trigonometric functions

How big is the slope at a 30 ° angle?

tan 30 ° = a / b

b is the horizontal distance. We'll set it 100m.

So a (height difference) = tan 30 ° * 100 = 57.74%

The pitch of a garage roof should be 15 °. The roof should have the greatest height on the house wall. The width of the garage is 7m. If the garage is supposed to be at least 2.5m inside, how high is the roof at its highest point?

Solution and solution:

tan α = opposite side / adjacent side

tan 15 ° = x / 7m

x = tan 15 ° 7m = 1.88m

The roof has a highest point of 8.88m

**Related topics**

Linear function

Coordinate system

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