Initial Publication Date: July 19, 2011

# Calculating slope

*Practice problems*

## Avalanche hazards

**The following questions relate to the map below. You can view a larger version of this map by clicking on the map. You can also download a sheet that has the map and the sample problems (Acrobat (PDF) 83kB Jul25 09) so you can print it and try them on your own.**
You have recently purchased land around Pioneer Ridge (on the map above). You plan to build a ski resort and need to decide where to build your lodge. You have chosen two sites that have the aspects you want for your lodge (a view of the river and proximity to some nice steep (double diamond) slopes. You know that avalanches are most likely to happen on slopes that are 60-100%, so you need to know which of the sites will be less likely to be buried in an avalanche. Use the information on the map to calculate % slope and then decide which site is more realistic for your lodge.

This question is asking you to calculate % slope. So, let's remember the steps for calculating percent slope.

- First, we need to get comfortable with the features of the topographic map of interest.
- What is the contour interval (sometimes abbreviated CI)?
The contour interval on this map is 80 ft.

- What is the scale of the map?
The scale is shown in the lower left. It is a graphical scale so you'll have to get out your ruler!

- What is the feature for which you want to know the slope?
For this problem, there are actually two hill slopes of interest. They are marked in red on the map.

- First, you need to know "rise" for the features. "Rise" is the difference in elevation from the top to bottom (see the image above).So determine the elevation of the top of the hill (or slope, or water table).
The top of the red line for site A is at 4400 ft. The bottom is at the 2400 contour line. The difference in elevation (rise) is the top minus the bottom (4400 ft - 2400 ft = 2000ft). The top of the red line for site B is also on the 4400 ft contour line. Line B's bottom is at the 2400 contour line. The difference in elevation is the top minus the bottom (4400 ft - 2400 ft = 2000), so **"rise" = 2000 ft for both lines**.

- Next you need to know"run" for the features. "Run" is the horizontal distance from the highest elevation to the lowest. So, get out your ruler and measure that distance.
The red line above site A is about 1.1 miles long. The line above site B is 0.5 miles long.

- To calculate
**percent slope**, both rise and run must be **in the same units**(for example, feet or meters). In this problem, the elevation is in feet but the scale is in miles. So we'll need to convert the horizontal distance to feet.
Right now you have rise in feet and run in miles. Let's convert the miles to feet by multiplying by the appropriate conversion factor: 1 mile = 5280 feet. So, we should multiply "run" by

`\frac{5280\ ft}{1\ mi}` for each of the sites:

- For site A: `1.1\ mi\times\frac{5280\ ft}{1\ mi}=5808\ ft`
- For site B: `0.5\ mi\times\frac{5280\ ft}{1\ mi}=2640\ ft`

- Now that we have converted so that both elevation and distance have the same units, we can write an equation for slope: rise over run(implying rise divided by run).
We know that for site A, rise is 2000 ft and run is 5808 ft; for site B, rise is 2000 ft and run is 2640 ft:

- For site A: `\frac{rise}{run}=\frac{2000\ ft}{5808\ ft}`
- For site B: `\frac{rise}{run}=\frac{2000\ ft}{2640\ ft}`

- To get to % we need to multiply the calculated slope (which is unitless because the units cancel on the top and bottom) by 100% so that our equation looks like this:

`\frac{text{rise}}{text{run}}=text{slope}`

Start with rise over run and multiply by 100%:

- For site A: `\frac{rise}{run}\times 100%=\frac{2000\ ft}{5808\ ft}\times 100%=%\ text{slope}`
- For site B: `\frac{rise}{run}\times 100%=\frac{2000\ ft}{2640\ ft}\times 100%=%\ text{slope}`

- Now plug in your numbers and calculate % slope!
- For site A: `\frac{rise}{run}\times 100%=\frac{2000\ ft}{5808\ ft}\times 100%=34%\ text{slope}`
- For site B: `\frac{rise}{run}\times 100%=\frac{2000\ ft}{2640\ ft}\times 100%=76%\ text{slope}`

## Water Table

You are working in an area with an important aquifer that has been contaminated by a buried tank. You need to know the slope of the water table so that you can calculate how quickly the contaminants will get to the nearby wells. You measure the depth to water in two wells: Well A has water at 649 m elevation. Well B has water at 937 m elevation. The two wells are 0.7 km apart. What is the slope of the water table (in m/m)?

In this case, you are given the elevations rather than a map so you don't need to worry about reading a map. Instead we start with the second step.

- First, you need to know "rise" for the features. "Rise" is the difference in elevation in the wells (the top of the water table).
The highest elevation is 937 m and the lowest elevation is 649 m. So the difference is **937 m - 649 m = 288 m** so **"rise" = 288 m**

- Next you need to know"run" for the features. "Run" is the horizontal distance from the highest elevation to the lowest.
In this case run is given to you and it is in km. **Run for this problem is 0.7 km**

- This problem is asking you to calculate slope in m/m. How do you do that?
- Right now you have rise in meters and run in km. So, first you need to convert "run" or the horizontal distance to m. To do this, we multiply by:

`\frac{1000\ m}{1\ km}`

So, we have `0.7\ km\times\frac{1000\ m}{1\ km}=700\ m`

Note that the units cancel out so that we are left with meters.

- Remember that slope is "rise over run". The phrase "rise over run" implies that you will need to divide. The equation for slope looks like this:
`\frac{text{rise}}{text{run}}=text{slope}`

- Now, take the difference in elevation and divide it by the horizontal difference (always making sure you keep track of units).
In the well problem above, rise = 288 m and run is 700 m. So we set up the problem like this:

`\frac{text{rise}}{text{run}}=\frac{288\ m}{700\ m}=text{slope}`

- Finish the calculation using your calculator (or doing the calculations by hand).
Now we just divide the rise by the run and wind up with:

`\frac{text{rise}}{text{run}}=\frac{288\ m}{700\ m}=0.41\ \frac{m}{m}`

You end up with 0.41 m/m which you can then use to figure out the velocity of the contaminants (but that's another equation all together and you don't need to do it here).

## Next Steps

If you feel comfortable with calculating slopes, you can go on to the assessment.

**Remember, you can only take this assessment once, so make sure you're ready!**