've been working on this problem with Mathematica and by hand-help with either would be fantastic. The blade of grass is given by the line segment from (32,1/5) and (32,8). The 2D hill is given by f(x)=(1/16 x^2 - 2 x + 80)/(1/16 x^2 - 2 x + 20)^2
I am struggling to figure out how to set the equations so that I can solve the point on the hill line, especially because I have to qualify that the other point on the line (which slope equal to f'(x)) must intersect only between y=1/5-8 AND not interest h(x) again (which defines the hill). Any help, detailed or theoretical is much appreciated!
Let's say the point $x$ is the point on the hill where the ant can see the top of the blade of grass. Plugging $x$ into $h(x)$ you can get the $y$ value.
Then you can set up an equation for the slope of $h(x)$ at $x$ using the derivative of $h(x)$ and an equation of the line from that point to the top of the blade of grass using $\frac{y-y}{x-x}$ as its slope. And these two slopes should be equal.
EDIT: to clarify, i think the line equation should be the line that goes through our $(x,y)$ on the hill, and the point $(32,8)$. I am imagining a big arm hinged at one end at $(32,8)$ and allowed to drop onto the hill. And from the characteristics of the $h(x)$ I think there is only one point in the domain we are interested in that will have the same slope as that line will, since it is decreasing over that domain.