Section 11.3 problem 11

I'm stumped on how exactly to get this answer.  I just can't see the
relation between Fy and the contour lines.

The notion is that the partial derivative is the limit of the difference quotient 

this means that when h is small, the partial derivative is approximately equal to the finite difference quotient:

This is is an important and useful fact for an engineer, because a lot of times the only thing you know about the function is some contours or some data points.  For this problem you can start at the point (1,3), where you're right on the contour for 10, so you can get f(x,y)=10, and if you increment y by about h=2.6 you get the  f(x,y+h)=8

--that's all the information you need to get the approximation