MathsJam Jars

9694 /www/nrich/html/content/id/9694/ 1 2012-12-05T12:02:21 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> A series of jam jars of uniform cross section look like letters when viewed face-on. They are 1cm thick, and the corners of the vessels have either whole or half cm values for their coordinates. <mdo:image alt="" height="300" src="6424.jpg" width="537"></mdo:image> Hot, smooth jam is poured slowly into each vessel through one of the holes at the top at a rate of 1 cm$^3$ per second. Seven of the jam jars take the same time to fill up. Which are they? Which one takes the longest to fill? Which would fill up first? The height of the jam in one of the vessels is measured and a chart of the height against time is plotted, as follows: <mdo:image src="jampic.jpg"></mdo:image> Which jam jars does this chart correspond to? Can you explain what each part of the chart corresponds to? The very observant jam maker might have noticed that the chart given is actually slightly inaccurate: at certain points it is an over-measurement and at certain points it is an under-measurement. Can you see where and why? Make charts of height against time for some of the other letters. p.s. If you feel the need to question the 'runnyness' of the jam, we can assume that it is of low viscosity, contains no 'bits' and remains at the same temperature throughout. You can decide on the flavour. You might also like thinking about <a href="http://nrich.maths.org/6512">this problem</a>. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">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. <div style="text-align: center;"> <mdo:image height="192" width="154" src="MathsFiller.png" alt="Letter M"></mdo:image></div> <div style="margin-left: 40px;">1. 0 - 3 minutes - filling one 'leg' of M, with rate of height increase constant due to constant width</div> <div style="margin-left: 40px;">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</div> <div style="margin-left: 40px;">3. 7 - 11 minutes - water fills top rectangular section with constant rate of height increase</div> <div style="margin-left: 40px;">4. 11 - 14 minutes - filling up top two trapezoidal sections of M</div> <div style="margin-left: 40px;">5. 14 - 16 minutes - letter completely full - no further height gain</div> <div>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. </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">You can find Teachers' Notes for this problem <a href="http://nrich.maths.org/6424/note">here</a>. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Does it help to work out the cross sectional area of each letter? Can you picture how full each letter would be if one cubic centimetre of jam was poured in? What about two cubic centimetres? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">The V is 13 cm3 The A is 12 cm3 The S is 14.5 cm3 The others are all 14cm3 By calculating the cross-sectional area of the shapes (counting squares), the following volumes can be obtained: <table border="1" style="margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">Letter</td> <td style="text-align: center;">Volume $\textrm{cm}^3$</td> </tr> <tr> <td style="text-align: center;">I</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">L</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">O</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">V</td> <td style="text-align: center;">13</td> </tr> <tr> <td style="text-align: center;">E</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">M</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">A</td> <td style="text-align: center;">13.5</td> </tr> <tr> <td style="text-align: center;">T</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">H</td> <td style="text-align: center;">14</td> </tr> <tr> <td style="text-align: center;">S</td> <td style="text-align: center;">14.5</td> </tr> </tbody> </table> Therefore {I, L, O, E, M, T, H} all take the same time to fill. S takes the longest and V the shortest time. The chart corresponds to the letter M (see diagram below for numbering): 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 $4 \textrm{ cm}^3$ 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 <div style="text-align: center;"><mdo:image height="192" width="154" src="MathsFiller.png" alt="Annotated Letter M" style="margin-left: auto; margin-right: auto;"></mdo:image></div> <div>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.</div> </mdoxml> 2 3 0 0 0 1 0 MathsJam Jars Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like? Sequences, Functions and Graphs Graphs