On topographical maps, each closed figure represents a fixed...

On topographical maps, each closed figure represents a fixed elevation (a vertical change) according to a given contour interval. The measured distance on the map from point $A$ to point $B$ indicates the horizontal distance or the horizontal change between point $A$ and a location directly beneath point $B$, according to a given scale of distances. (FIGURE CAN'T COPY) As part of park maintenance, the 2 by 4 handrail alongside a mountain trail leading to the summit of Mount Marilyn must be replaced. In the figure, the contour interval is 1: 200 (each figure indicates a change of $200 \mathrm{m} \text { in elevation })$ and the scale of distances is $1 \mathrm{cm}=$ $400 \mathrm{m} .$ (a) Find the change of elevation from $A$ to $B ;$ (b) use a proportion to find the horizontal distance between $A$ and $B$ if the measured distance on the map is $4.33 \mathrm{cm} ;$ and $(\mathrm{c})$ draw the corresponding right triangle and use a special triangle relationship to find the length needed to replace the handrail (recall that $\sqrt{3} \approx 1.732$.

On topographical maps, each closed figure represents a fixed elevation (a vertical change) according to a given contour interval. The measured distance on the map from point $A$ to point $B$ indicates the horizontal distance or the horizontal change between point $A$ and a location directly beneath point $B$, according to a given scale of distances. (FIGURE CAN'T COPY)
As part of park maintenance, the 2 by 4 handrail alongside a mountain trail leading to the summit of Mount Marilyn must be replaced. In the figure, the contour interval is 1: 200 (each figure indicates a change of $200 \mathrm{m} \text { in elevation })$ and the scale of distances is $1 \mathrm{cm}=$ $400 \mathrm{m} .$ (a) Find the change of elevation from $A$ to $B ;$ (b) use a proportion to find the horizontal distance between $A$ and $B$ if the measured distance on the map is $4.33 \mathrm{cm} ;$ and $(\mathrm{c})$ draw the corresponding right triangle and use a special triangle relationship to find the length needed to replace the handrail (recall that $\sqrt{3} \approx 1.732$.

Key Concepts

Topographical Maps and Contour Lines

This concept involves understanding that topographical maps show elevation using contour lines, where each closed curve represents a constant elevation. The spacing and number of these lines help determine the relative steepness and shape of the terrain, making them an essential tool for interpreting how the landscape rises or falls.

Map Scale and Proportional Reasoning

A map scale provides a ratio that converts measurements on the map to real-world distances. This concept is crucial for applying proportional reasoning, where you multiply the measured distance on the map by the scale factor in order to obtain the actual horizontal distance between two points.

Right Triangle Relationship in Elevation Change

When dealing with elevation changes and horizontal distances, a right triangle is often formed with the vertical leg representing the change in elevation and the horizontal leg representing the measured ground distance. This geometric representation facilitates the calculation of the actual slope distance along the terrain using the Pythagorean theorem or trigonometric relationships.

Special Right Triangles

Special right triangles, such as the 30-60-90 triangle, have fixed ratios between their sides. Recognizing such triangles in practical problems can simplify the computation of unknown lengths. These geometric relationships can be applied to quickly determine the hypotenuse or other unknown side when one side and the angle or ratio is known.

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