One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce).  The postage for a letter weighing $4.5$ ounces was
$\text{(A)}\ \text{96 cents} \qquad \text{(B)}\ \text{1.07 dollars} \qquad \text{(C)}\ \text{1.18 dollars} \qquad \text{(D)}\ \text{1.20 dollars} \qquad \text{(E)}\ \text{1.40 dollars}$
After the first ounce, there are $3.5$ ounces left.  Since each additional ounce or fraction of an ounce adds $22$ cents to the total cost, we need to add $4\times 22$ to the cost for the first ounce.
So, the total price is $30+4\times 22=118$ cents. The answer is choice $\boxed{\textbf{(C)}~\mathbf{1.18}~\textbf{dollars}}$.