# Find the average rate of change between two points on a contour map

Disclaimer: This a homework question for a multivariable calculus course.

The problem:

Find the average rate of change between $A$ and $C$ using the given contour map.

The average rate of change for a contour map is given by $\frac{\Delta altitude}{\Delta horizontal}$.

What I've done:
$\Delta altitude = -9 - (-3) = -6$.
$\Delta horizontal = \sqrt{(6-2)^2 + (5-4)^2} = \sqrt{17}$

Therefore, the average rate of change between $A$ and $C$ should be $\frac{-6}{\sqrt{17}}$. However, according to the answer key, the average rate of change is $\frac{-9-(-3)}{\sqrt{2^2 + 1^2}} = \frac{-6}{\sqrt{5}}$.

Did I do something wrong along the way, or is the answer key wrong?

• My guess would be the answer key is wrong, based under the assumption that the 5 they got was from failing to square the change in x and change in y's, which would lead to 4+1 instead of 16+1
– Alan
Sep 20, 2016 at 5:09
• Actually, in the answer key, they show sqrt(2^2 + 1^2) for the horizontal. Updated the question to clarify.
– null
Sep 20, 2016 at 5:18

$\Delta altitude = -9 - (-3) = -6$.
$\Delta horizontal = \sqrt{(6-2)^2 + (5-4)^2} = \sqrt{17}$
$\frac{\Delta altitude}{\Delta horizontal}=\frac{-6}{\sqrt{17}}=\frac{-6\sqrt{17}}{17}$.