Question on profit maximization with government taxation (I) A monopolist has a cost function $c(q)=q$. It faces the following demand function $ q(p)=100/p$ for $p\le 20$ and $q(p)=0$ for $p\ge 20$. What are the profit maximizing price and output. 

(ii) a monopolist has a cost function $c(q)=\alpha q$ where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is $-3$. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?

My solutions are as follows:

I am not sure whether it is correct or not. Please tell me and show me my mistakes.
ii.

I don't think the demand function in (i.) applies for this question. Here's what I did:

By Lerner Index, $\displaystyle{\frac{P - MC}{P} = \frac{-1}{E_d}}$

Since $E_d = -3$, we get $P = \frac{3}{2} MC$

We know before tax MC is $c'(q) = \alpha$, and after tax MC is $\alpha + 6$

In this sense, MC increased by 6, so P will probably increase by 9