
We want to find
such that
. This means



Integrating both sides of the latter equation with respect to
tells us

and differentiating with respect to
gives

Integrating both sides with respect to
gives

Then

and differentiating both sides with respect to
gives

So the scalar potential function is

By the fundamental theorem of calculus, the work done by
along any path depends only on the endpoints of that path. In particular, the work done over the line segment (call it
) in part (a) is

and
does the same amount of work over both of the other paths.
In part (b), I don't know what is meant by "df/dt for F"...
In part (c), you're asked to find the work over the 2 parts (call them
and
) of the given path. Using the fundamental theorem makes this trivial:


I believe this is scale factor? SF=new/old. 10/8= a scale factor of 1.25 (the image gets 1.25x bigger). To find x, multiply 5 by 1.25 and subtract the original length of 5 (you subtract this length because x is only the value of the new part of the triangle). X=6.25-5, X=1.25. Try to find y yourself using this scale factor. (Brainliest would be appreciated)
The square root of 81 is 9
Your answer is <em>9
</em>Work:
9 x 9 = 81