Inclines - mathematics activities

Description

This photograph of a road sign suggests investigations of the different indicators of slope used in our environment. It may lead you to explore trigonometric relationships in relation to gradient.

Activities

1. The sign in the photograph indicates that there is a steep hill on the road ahead in this rural scene. Just what, precisely, is the sign 'saying'?
2. What is the measure of the smallest angle of the black triangle (assuming it is a right-angled triangle) shown on the sign? What ambiguities might occur here?
3. Use an internet search engine to find a different photograph indicating a steep hill on a road ahead. What difference is there, if any, between the information given in the road sign you chose and the one here? Which is indicating the steeper incline?
4. Draw a mathematically accurate diagram that compares the maximum slopes of a road (1 in 6), a narrow gauge railway track (1 in 30) and a main line railway track (1 in 40). How would each of these maximum slopes be shown as percentages? Include these percentages in your diagram.
5. What does your diagram for question 4 say about the relative abilities of cars and trains to cope with inclines? Would cyclists prefer to ride on cycle paths at the side of roads or on paths made on disused railways? Support your response with a mathematical explanation.
6. Railway slope indicators are usually given as a ratio. Is this ratio different from that used for roadways?
7. Investigate 'scenic railways' or 'steep streets'. Report on your investigation, including a discussion about the slope or gradients met in your work. Diagrams would be a good way to explain key ideas.
8. Construct a table that compares the angle of slope to the horizontal, the ratio gradient, and the percentage slope for at least five different real slopes in our environment (eg ramps for the disabled, scenic railways, roads, tram lines, bike paths, etc). Justify your calculations in each case. You might be able to construct a computer program that automatically converts from one method of stating slope to another. If so, demonstrate how your program works.