Answer
(a) $T=59.82^{\circ}C$
(b) $T=19.55^{\circ}C$
Work Step by Step
We write simple proportions to find the answer. We know that thermal expansion is directly linearly proportional to heat, so we write the proportions as follows, where $h =$ the height of the alcohol at a certain temperature and $T =$ temperature.
For a single material, the same amount of heat will lead to a certain change in height, which means that this proportion is constant. We call this constant the linear thermal expansion coefficient, which is unique to every material.
$\frac{h_{f}-h_{i}}{T_{f}-T_{i}}=\frac{\Delta h}{\Delta t} =$ the coefficient of linear thermal expansion
First, we must find the constant coefficient.
$$\frac{22.79cm-12.61cm}{100.0^{\circ}C-0.0^{\circ}C}=.1018 cm/^{\circ}C$$
Then, we use that coefficient to solve parts (a) and (b).
(a)$$.1018 cm/^{\circ}C=\frac{18.70cm-12.61cm}{T_{f}-0.0^{\circ}C}$$
$$T_{f}=59.82^{\circ}C$$
likewise, (b) $$.1018 cm/^{\circ}C=\frac{14.60cm-12.61cm}{T_{f}-0.0^{\circ}C}$$
$$T_{f}=19.55^{\circ}C$$