Maths filler 2
Problem
This problem follows on from the problem Maths Filler 1, although it can be attempted independently from this.
A vessel is constructed from a connected sequence of block-letters (assume that corners of the vessel lie either on grid vertices or half way between grid vertices). Water is poured slowly into the hole on the left (marked in blue) at a rate of 1cm$^3$ per minute.
A scale on the left of the vessel measures the height $H(t)$ of the pool of fluid which forms.
How long will it take to fill the vessel to reach each of the markers on the left hand side?
Use this to plot an approximate height-time chart.
Extensions: Think about the issues involved in drawing a completely accurate height-time chart.
How would the chart change if we were to fill through a different hole?
Getting Started
The A is the most awkard letter. Before you start plot the A on squared paper and work out the areas of each of the cross sections at the various heights. Don't forget that you can work in fractions of whole squares and use the formula for the area of a trapezium.
Student Solutions
The first 6 marks are filled one every half minute.
Then for mark 7 we first fill up to 6 completely:
$M+ = 1+3+1 = 5$
$A+ = \frac{3.5+4}{2} \times 2 +4 - \frac{1+2}{2} \times 2 +2 = 10.5$
$T+ = 6+2 = 8$
$H+ = 3+3+3+1 = 10$
$S = 4+1+1+1+1+0.5 = 8.5$
42 total
And then fill up to 7:
$M = 2$
$A = \frac{3.5 + 3.375}{2} \times 0.5 -1 \times 0.5 = \frac{39}{32}$
$T = 2$
$H = 2$
$S = 2 \times \frac{1}{2} + \frac{1}{8} = \frac{9}{8}$
$\frac{267}{32}$ total (sum to this point = $\frac{1707}{32}$)
And following the same process for the rest of the levels:
Level 8: $\frac{225}{32}$ total (sum to this point = $\frac{1932}{32}$)
level 9: $\frac{267}{32}$ total (sum to this point = $\frac{2199}{32}$)
Level 10: $\frac{249}{32}$ total (sum to this point = 76.5)
The sum of all these is 76.5.
Now if we check the total size to check it's the same: $24 \times 5 - 23.5 \textrm{ spaces} = 76.5$ check!
So we can plot these points on a graph! Joining them up is another matter...