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Step 1: To understand the difference between a valley and a canyon, it is essential to define each term.
A valley is a low area between hills or mountains, typically with a river flowing through it. Mathematically, a valley can be thought of as a concave surface in the landscape. In terms of cross-sectional shape, a valley usually has gentle, sloping sides that gradually converge towards the riverbed. This shape can be represented as a concave up function or curve.
Step 2: On the other hand, a canyon is a deep, narrow valley with steep sides, usually formed by the erosive action of a river over a long period.
Canyons often have a more dramatic cross-sectional shape compared to valleys, with nearly vertical walls and a narrower floor. This shape can be represented as a significantly steeper concave up function or curve.
Final Answer
In summary, a valley is a broader, shallower, and gentler low area between hills or mountains with a river flowing through it, while a canyon is a more dramatic, deeper, and narrower feature with near-vertical walls formed by the erosive action of a river over time.
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