Well done to Anurag and Christina for their
solutions to this problem:

The seven letters that take the same time to fill up are: I,L,O,E,M,T and H, all with a volume $14$cm$^3$ and thus taking $14$ minutes to fill.

The letter S takes the longest to fill up ($14.5$cm$^3$, $14.5$ minutes to fill).

The letter V fills up first ($13$cm$^3$, $13$ minutes to fill).

The letter A will take $13.5$ minutes to fill.

The graph corresponds to the letter M. Some points in the graph are over measured, for instance, the points between 4 and 5 cm.

The seven letters that take the same time to fill up are: I,L,O,E,M,T and H, all with a volume $14$cm$^3$ and thus taking $14$ minutes to fill.

The letter S takes the longest to fill up ($14.5$cm$^3$, $14.5$ minutes to fill).

The letter V fills up first ($13$cm$^3$, $13$ minutes to fill).

The letter A will take $13.5$ minutes to fill.

The graph corresponds to the letter M. Some points in the graph are over measured, for instance, the points between 4 and 5 cm.

1. 0 - 3 minutes - filling one
'leg' of M, with rate of height increase constant due to constant
width

2. 3 - 7 minutes - further water
will run over in to the central dip of the M, and then once this is
filled into the opposite leg. These have a combined volume of
$4$cm$^3$ and so take 4 minutes to fill

3. 7 - 11 minutes - water fills top
rectangular section with constant rate of height increase

4. 11 - 14 minutes - filling up top
two trapezoidal sections of M

5. 14 - 16 minutes - letter
completely full - no further height gain

Section 4 in the period 11-14 minutes should actually be
represented by a curved line on the chart as the width of the
section being filled is changing with height, and therefore so will
the rate of height increase.