Question

Everything being equal, including speed and mass, Like a man walking up a slope, does the amount of energy expended on say a 10% inclined slope double with a 20% inclined slope? if travelling at the same speed and everything else the same?
In other words, is energy expended related arithmetically and constant according to the slope of the incline? and resistance, or is it proportional to the slope by some other factor, geometric increase?

Answer 1

His potential energy is E=mgy
where y is his height above the bottom of the hill and
g=9.8 m/s2 is the acceleration due to
gravity. The rate at which his energy is changing is

$\frac{dE}{dt}=P=mg\frac{dy}{dt}=mgvy=mgvsin$

so this rate (called power) depends only on the vertical, not the horizontal component of his velocity. Now, you have
made things difficult for me because you have not specified
the angle, but rather the inclination which is actually the tangent

of the angle (rise/run) times 100%. I will call this inclination
the grade G=100tan. For example,
let's take a man with a mass 100 kg walking with a
speed of 2 m/s up the 10% and 20% inclines you to specify. The angles
would be 10%=tan-1(10/100)=5.710
and 20%=tan-1(20/100)=11.310.
So the power values are P10%=100x9.8x2xsin(5.71)=195.0
Watts and P20%=100x9.8x2xsin(11.31)=384.4
Watts. The ratio of these two is 1.97, close to, but not
equal to, 2.0; for small angles the sine and tangent are
approximately linear functions but as the angle (hence
inclination or grade) increases this is no longer a good
approximation. The graph shows the power for grades up to
200% (about 63.40); as you can see, the power
increases approximately linearly only up to about a grade of
about 30% which is about 170.