The Craftsman's Questionable Son This activity requires students to measure the volume of objects using water displacement, the mass of objects using gram scales, to calculate the density of objects, and to solve a story problem using either graphical or algebraic techniques. The graphical techniques are the easiest but provide the least accurate answer. The algebraic solution is the most accurate but relies on students' ability to manipulate equation. An intermediate stage allows students to solve the problem using algebraic techniques but without sophisticated equation manipulation skills. Students should have been introduced to the idea of density and how it is a unique and unalterable characteristic of a material. If the students are not familiar with volume measurement by displacement, then Archimedes's Bathtub may be suitable reading. Materials: gram scales, apparatus for measuring water displacement, 1 'gold' crown, 1 'alloy' crown, and at least one 'commissioned' crown per student group. The crowns to use are a combination of heavy steel bolts, nuts, and washers, and nylon spacers. Create the 'gold' crowns with just the hardware; 'alloy' crowns with nylon spacers tied together with fishing line, and the 'commissioned' crowns with hardware and nylon nylon spacers intermixed. The nylon spacers are very light and require accurate gram scales in the classroom. Obviously, other materials may be used as well. The most important factor is that the students can measure the gram weight of the crowns to enough accuracy to reliably measure different densities for the 'gold' and 'alloy' crowns. The more resolution your classroom scales have, the closer together the densities of the 'gold' and 'alloy' crowns may be. You will definetly want to run a quick test of the crown material and the scales that will be used to be sure the densities can be determined with sufficient accuracy.
The description of Archimedes's New Problem may be assigned as reading to the students or simply described by the instructor. For a real hoot, you can use the provided script and have two young thespians act a skit out. In this case, I recommend having Archimedes and the King use the mock crowns concocted for the lab as if they were the real thing. If Archimedes's New Problem is simply described, the following facts must be presented: Archimedes has a pure gold crown Archimedes has a crown made entirely of the 'alloy' without any gold Archimedes has several other crowns that may be made of varying proportions of gold and alloy Archimedes knows how to measure the volume of objects precisely (the same way students will, by water displacement) Archimedes can measure the mass of objects (the students will use gram scales for this) Archimedes only has to determine how much gold (the mass) is in the commissioned crowns Three different techniques may be used to determine the mass of gold in the commissioned crowns. The first technique is graphically based and should be accessible to any student familiar with graphing tables of numbers on a coordinate grid in order to plot lines. The graphical technique results in an answer with the most error since the solution is simply estimated from a graph. The second technique uses the same theoretical approach as the first but replaces the graphing with algebraic manipulation of linear equations. The technique provides the most accurate answer but requires students which are comfortable with algebraic manipulation and variable substitution in equations. The third technique develops the same initial equations used for the pure algebraic approach, but allows the immediate substitution of known quantities and thereby simplifies the algebraic manipulations into solution patterns that students have probably seen before. Students still need to be familiar with solving a one variable equation.
Graphic Solution Perhaps the easiest physical constraint to describe in the problem is the volume constraint. Since the crowns are merely a physical combination of gold and Guido's alloy, then there is a specific volume of gold in the crowns and a specific volume of allow in the crowns. These two volumes, when added together, must equal the total volume of the commissioned crowns: Volume of Gold Volume of Alloy=Total Volume The students can measure the total volume of the commissioned coins. Suppose the total volume is 36ml. Then they can create a table and fill in the red cells: Volume of Gold Volume of Alloy Total Volume 0ml 10ml 0ml 10ml The data points of this table can be graphed with the Volume of Gold as the Y Axis and the Volume of Alloy as the X Axis, connect these points with a straight line. This line represents all the combinations of gold and alloy volumes which sum to 36ml, the volume of our commissioned coins. 36ml 36ml 36ml 36ml Next, the students can measure the mass of the commissioned coins, and find the density of pure gold crown and the pure alloy crown. Although students can calculate the density of the commissioned crowns, the value won't be needed to find the solution. Suppose the students determine the mass of the commissioned crowns to be 96.39g, the density of the gold to be 7.796g/ml and the density of the pure alloy to be 1.155g/ml. Imagine now that the commissioned crowns were made from 100% gold and their mass was 96.39g what would the volume of these crowns be?
Volume of Gold= mass density = 96.39 7.796 =12.364 The volume would be 12.364ml, much less than the volume of our commissioned crowns! Now suppose the commissioned crowns had a mass of 96.39g and were forged entirely of alloy; what would their volume be? Volume of Alloy= mass density = 96.39 1.155 =83.45 Then the volume would be 83.45ml, much more than the volume of our commissioned crowns! Let's plot these points. The first point is (0,12.364) representing 0ml of alloy and 12.364ml of gold. This is one way to get a mass of 96.39g. The second point will be (83.45,0) representing 83.45ml of alloy and 0ml of gold. This is another way to get a mass of 96.39g. Connect these two points on the graph with a straight line (use a ruler and be very accurate). Is it possible that all the other points on this line represent the different combinations of gold and alloy volumes that yield commissioned crowns with mass 96.39g? Let's select a couple of points on this graph and test this hypothesis. Fill in the green squares below by using the measured densities of the pure gold and pure alloy and determine the total mass. Point Volume of Alloy Mass of Alloy Volume of Gold A (20,9.5) 20ml (20)(1.155)=23.1g 9.5ml Mass of Gold B (60,3.5) 60ml 3.5ml (3.5)(7.796)=27.286 Total Mass
It certainly looks like this new line represents all the possible combinations of gold volume and alloy volume which would create commissioned crowns of mass 96.39g. We are trying to find the amount of gold in the commissioned crowns, and we know two properties that the commissioned crowns must have. The total volume must be 36ml and the total mass must be 96.39g. We can draw two lines representing these constraints: the red plot is the volume constraint, the green line is the mass constraint. What happens if we plot these two lines on the same graph? EUREKA! These lines cross at only one point! Only the combination of 27.7ml of alloy and 8.3ml of gold adds up to 36ml of total volume and 96.39g of mass! The commissioned crowns have 8.3ml of gold, so the mass of the gold in the commissioned crowns is: (8.3ml)(7.796g/ml) = 64.7g
Graphic Solution Worksheet 1 2 3 Measure the total volume of the commissioned crowns Plot the line representing all the gold and alloy volumes that sum to the total volume measured in step #1 Measure the total mass of the commissioned crowns 4 Measure the density of the pure gold crown. 5 Measure the density of the pure alloy crown 6 7 8 9 Plot the line representing all the gold and alloy volumes that would make commissioned crowns whose mass is the same as measured in step #3 Estimate the point of intersection of the two lines drawn in steps #2 and #6 Check your work by calculating the mass of gold and alloy in the commissioned crowns Take the commissioned crowns apart and check the values you calculated!
Complete Algebraic Solution The students will be able to measure 3 different physical quantities: 1. the density of the pure gold crown 2. the density of the pure allow crown 3. and the volume of the commissioned crowns Although students can measure the density of the commissioned crowns, the value is not required to determine the solution. There are two physical constraints in the problem. Since the crowns are merely a physical combination of gold and Guido's alloy, then there is a specific volume of gold in the crowns ( ), and a specific volume of allow in the crowns (V a ). These two volumes, when added together, must equal the total volume of the commissioned crowns (V). In other words: V a =V The students can only measure one of these volumes, which is the total volume of the commissioned crowns, (V). Finally, since the combination of the gold and alloy do not create a chemical reaction, then the mass of the gold in the commissioned crowns (m g ) added to the mass of the alloy in the crowns (m a ) must equal the total mass (m) of the commissioned crowns. m g m a =m The students can only measure one of these masses as well, the total mass of the gold and alloy together, in other words the total mass of the commissioned crowns (m). Now let d g represent the density of gold and d a the density of the alloy. The mass equation above can be rewritten in terms of and V a : d g V a d a =m The critical logic is noticing that we can invoke a change of variables or substitution using the first volume equation: the volume of alloy is equal to the total volume less the volume of gold: V a =V d g V d a =m d g V d a d a =m At this point the equation has only one unknown variable ( ), the students have measured the densities of gold and the alloy as well as the total volume and mass of the commissioned crowns. These measured values can be substituted into the equation and the volume of gold can be solved for. Below is a continued derivation that provides an analytical solution for. d g d a V d a =m d g d a =m V d a = m V d a d g d a
Here is an example calculation from 1/4 x 2 1/2 hardware bolts and nylon spacers (available at most hardware stores). I used a gram scale with about 1/2 gram accuracy and large graduated cylinder with about 2 ml accuracy. The 'gold' had a measured density of 7.796 g/ml and the 'alloy' had a density of 1.155 g/ml. The 'commissioned crown' had a volume of 36ml and a mass of 96.39g. Using = w V d a d g d a the volume of 'gold' was calculated to be 8.27ml and the mass of 'gold' was then determined to be 64.5g. The 'commissioned crown' was disassembled and the 'gold' was measured to be 65.2g with a volume of 9ml, the calculated volume and mass had % errors of 8.2% and 1.1% respectively.
Intermediate Solution The complete algebraic manipulation may be cut short at V a =V d g V a d a =m and measured values may be immediately substituted. When I tested this activity, the 'gold' had a measured density of 7.796 g/ml and the 'alloy' had a density of 1.155 g/ml. The 'commissioned crown' had a volume of 36ml and a mass of 96.39g. V a =36 7.796 1.155V a =96.39 Substituting V a = 36 into the second equation and solving for : 7.796 1.155 36 V a =96.39 7.796 41.580 1.155V a =96.39 6.641 =96.39 41.580 6.641 =54.81 = 54.81 6.641 =8.25 The volume of gold is 8.25ml. The mass of the gold can then be calculated to be (8.25ml)(7.796g/ml) = 65.2g
Hints The best results are obtained by measuring the mass and volume of the commissioned crowns all together. If the values are measured individually for each crown, and then added together, measurement errors will be compounded and the final solution may be much less accurate. At the end of the exercise students may want to know if their calculations are correct. By all means, let them take apart the commissioned crowns and measure the mass of the gold and alloy parts.
Archimedes's Bathtub Archimedes walked briskly to the King's estate after receiving a summons earlier that morning. His mind wandered back, 3 years prior, to when the King had last summoned him to the court. On that occasion, the King was concerned over a crown he had commissioned a local craftsman to make. The King had provided exactly enough gold for the crown but thought that the craftsman may have forged the crown out of a mixture of tin and gold keeping some of the gold for himself. Archimedes recalled that he had been troubled for days about how to precisely measure the volume of the ornate crown. He could determine the weight of the crown easily enough with a balance scale, and he knew that the density of gold was a constant value, but he couldn't measure the density of the crown without determining a very accurate measurement of the crown's volume. Archimedes chuckled to himself as he walked along the pathway to the King's court, reflecting that people always mocked him for thinking about his experiments day and night without rest, but on a particular night 3 years ago, his preoccupation with the King's crown provided a stroke of insight! He remembered that after a long unproductive day of thought and experiments with the questionable crown, he decided it would be best to take a nice, hot, relaxing bath. Archimedes, still deep in thought, had let his bathtub fill too full all the way up to the brim and he had barely managed to get the water turned off before the tub ran over and he had a serious mess on his hands. Grumbling to himself for not paying more attention, Archimedes resigned himself to mopping up his bathroom floor after his bath. Archimedes knew that as soon as he inserted himself into the tub, the hot water would spill over the edge of his bathtub and make a mess of his bathroom floor. So Archimedes slipped slowly into the tub, watching the water pour over the sides and onto the floor. Once settled, he looked around at the puddle he had made. It certainly seemed like a lot of water, and Archimedes began trying to estimate exactly how much cleanup he had in front of him. Would he need his small bucket, his medium bucket, or should he dig out his largest bucket when he began to mop up all the water? Curiosity got the best of Archimedes and he debated measuring how much water spilt out of the tub, because thoughts like that always came to the curious mind of Archimedes. And then, EUREKA! Archimedes realized that he already knew the volume of water on the floor, it was the volume of Archimedes' body! His legs and arms and torso couldn't occupy the same space in the tub as some of the water, so the water that would no longer fit spilt out onto the floor. He had displaced an amount of water equal to the volume of his own body. Archimedes cringed as he approached the gates to the King's estate. He remembered how the King had been pleased 3 years ago when Archimedes had proven that the craftsman had not stolen any of the King's gold. He also remembered how the King took Archimedes aside asking Please, don't go running through the streets of our City in the middle of the night yelling 'Eureka! Eureka!' no matter how important your next discovery turns out to be!
Archimedes's New Problem The King greeted Archimedes in his court with a furrowed, troubled brow. Archimedes, he said, I've got another problem with my crowns. The craftsman you proved honest 3 years ago has recently retired and left his trade in the hands of his son, Guido. Although I trusted the craftsman with many commissions after your density demonstration, I trust his son Guido about as far as I could throw him! Well, then good news I have for my Majesty today! boasted Archimedes, With the latest lever powered war machine I've designed you alone can wing Guido a good 300 to 400 cubits... The King raised his hand for silence. Enough of your levers and war machines, Archimedes. Today is not a day for levers and war machines, save these stories for the long winter nights around the hearth fire, or perhaps for when we are at war. Here is the problem I wish you to bend your mind around today, Archimedes. I commissioned Guido to make three different crowns and provided exactly enough gold for the forging. However, when Guido presented the finished pieces to me, he had gold left over! Guido claims that he has invented a new alloy, a mixture of gold and other metals. This new alloy is much better for crowns since it strengthens the metal against nicks and dings thereby decreasing the maintenance cost I have to pay to keep my crowns looking their finest. Here, said the King, holding up a very ugly dull specimen, is a crown Guido forged out of his new alloy but without any gold. It's a ruddy, ugly beast of a crown and I wouldn't be caught dead wearing it; but I've had all seven of my acrobats balance their entire weight on the thing, and had it dropped from the tallest towers in the City. The crown still doesn't have a scratch, dent, or nick in it so I guess Guido is on to something with this alloy of his. My problem, continued the King, is how can I be sure the amount of gold in the three crowns agrees with how much I provided and how much he returned to me? If the crowns contain 50% of my gold but Guido only returned 20% of what I provided, then he has stolen 30% of the gold he was given! And, as if that is not enough of a headache, Guido has told me he may have used different ratios of gold to the other metals based on the intended use of the crown. The crown I plan to wear day to day has more of the durable metals but isn't as shiny, the crown I had commissioned for special occasions has a higher percentage of gold to make sure its luster is noticed by everyone at our festivals. Archimedes, I want you to take these three crowns that Guido has forged and tell me how much gold is in them and I want to know precisely. If only 2 or 3 percent of the gold is missing I can chalk that up to an inefficient manufacturing processes, but if too much gold is missing, I'll have Guido banished from my Kingdom.
Archimedes thought for a moment and said, Your Majesty, this is an intriguing variation of the problem you presented to me 3 years ago, and I believe I can determine the exact amount of gold in the 3 crowns provided that Guido's new 'alloy' is merely a physical mixture of gold and other metals. Archimedes gathered 5 things from the court: the three crowns Guido had forged for the King; the dull, ugly crown Guido had presented to the King as proof of his alloy; and the same crown he had tested 3 years before which proved the father craftsman honest. Wait, Archimedes! shouted the King as Archimedes moved towards the exit. Don't you want to know how much gold I gave Guido and how much he returned to me? Archimedes turned and bowed reverently at the door, Your Majesty, those two facts won't help me determine how much gold is in the 3 new crowns. Those amounts will be used, along with the result of my calculations and measurements, to determine if Guido is an honest craftsman but that is simple arithmetic your accountants can perform. The sticky part of this problem is learning how much gold these 3 new crowns contain it is for this question alone that I must devise a solution.