Find r in terms of theta I have been given the following question:

My answer for question a) is $\theta/(2\pi)$. The books answer is $(2\pi-theta)/(2\pi)$.
I came to my answer by calculating the arc length of the sector which is theta.
Then I set up an equation making the arc length equal to the circumference of the circle hence $\theta = 2*\pi*r$.
 A: In response to a comment, we show how to solve d), maximizing the volume. 
We summarize calculations that you undoubtedly made. The circumference of the base of the cone is $2\pi -\theta$. (This was taken care of in a comment.)
So if the radius of the base is $r$ then $2\pi r=2\pi -\theta$, and therefore $$r=1-\frac{\theta}{2\pi}.$$
By the Pythagorean Theorem, the height $h$ of the cone is given by 
$$h=\sqrt{ 1-\left(1-\frac{\theta}{2\pi }\right)^2}.$$
The volume is $\frac{1}{3}\pi r^2h$. This is what we want to maximize. My typing fingers are tired, just imagine writing down an expression for volume in terms of $\theta$. We will use a method which bypasses that. 
We want to maximize the volume, or equivalently maximize $r^2h$, or equivalently maximize $(r^2)^2 h^2$.  Let $w=r^2$. We want to maximize $w^2(1-w)$, that is, $w^2-w^3$, where $0\le w\le 1$. 
This is a very easy calculus problem,   We get $w=\frac{2}{3}$.  But $w=\left( 1-\frac{\theta}{2\pi}              \right)^2$ and solving for $\theta$ is straightforward. We get
$$\theta=2\pi\left(1-\sqrt{2/3}\right).$$
Remark: The calculation could have been made by expressing volume explicitly as a function of $\theta$, and differentiating. However, the expression, though fundamentally simple, is rather daunting. At the very least one should make the substitution $t=1-\frac{\theta}{2\pi}$.
When we are maximizing/minimizing an expression $E$ that involves square roots, we can often simplify calculations by instead maximizing/minimizing $E^2$. 
