Answer
$11\text{ days}$
Work Step by Step
Let's note:
$x$=the number of days
$y$=the total cost
We build the system of equations:
$$\begin{cases}
y=x+121\\
y=12x.
\end{cases}$$
In order to find the value of $x$ so that the costs are equal, we graph the lines $y=x+121$ and $y=12x$ (see graph).
The intersection point appears to be $(11,132)$.
We check the solution algebraically:
$$\begin{align*}
y&=x+121\\
132&\stackrel{?}{=}11+121\\
132&=132\checkmark\\\\
y&=12x\\
132&\stackrel{?}{=}12(11)\\
132&=132\checkmark.
\end{align*}$$
The number of days after which the costs are equal is $x=11$.
If the daily cost for Option B increases to $12+a$, where $a>0$, the system of equations becomes:
$$\begin{cases}
y=x+121\\
y=(12+a)x.
\end{cases}$$
