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Learning Math Home
Measurement Session 1: Solutions
 
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A B C 
Homework

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Solutions for Session 1, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | B9 | B10 | B11 | B12 | B13


Problem B1

You could wrap tinfoil around the rock, just covering it; then unwrap the tinfoil, lay it flat, and measure its area. One reason to use this method is that it is far easier to measure a flat (two-dimensional) area than it is to measure the surface area of a three-dimensional object, and the tinfoil can be laid flat while still representing the three-dimensional surface area. Tinfoil is also quite flexible and can wrap tightly around most irregular surfaces of the rock.

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Problem B2

Yes, there is more than one choice. The units for surface area will be square: square inches, square centimeters, square millimeters, and so on. The size of the unit depends entirely on the size of the object being measured. Since the rock you used is relatively small, it is reasonable to use square centimeters or square half-centimeters as the unit of surface area.

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Problem B3

It is not very exact. There are several sources of error, including the error of estimating that the tinfoil has exactly enveloped the rock, and the error of rounding our answers. A closer approximation could be obtained by using a finer grid. It is more difficult to overcome the error of estimating using foil, but other materials could be used instead.

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Problem B4

Answers will vary, but you will probably find that the approximate areas are different for each type of grid. Again, measurement is not exact and is subject to error, including error introduced by the method of measurement itself.

But more importantly, you will probably notice that the inner and outer measures are closer together on the finer grid paper. This means that your answer will be closer to the actual area. By using smaller units, we can increase the precision of our measurements and get better approximations.

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Problem B5

The units are in milliliters. Yes, this unit can be used to measure the volume of a solid because of the relationship between a milliliter and a cubic centimeter. The number of milliliters of displacement will be equal to the number of cubic centimeters of volume for the rock.

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Problem B6

Probably not. Since rice is solid, it will not completely fill the container, and several measurements of the same amount of rice will likely have different results. Additionally, since we are measuring twice with two different methods, there is very likely to be a measurement error between the two.

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Problem B7

You can determine a measure (i.e., an approximation) for the weight of your rock relative to other known weights, but not the exact weight of the rock.

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Problem B8

The three-arm balance works on the lever principle, in which moving a weight farther from a balance point produces a greater force on that side of the balance. (This is the same principle used in balancing a seesaw.) We can determine an approximation of the rock's weight using this type of scale, but not the exact weight.

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Problem B9

Mass is a measure of the amount of material making up an object (specifically, its molecules). All objects have mass, but not all have weight, which is the effect of a gravitational field on a body that has mass. For example, a U.S. flag placed on the Moon has the same mass as one placed on the Earth, but it weighs less as a result of the Moon's gravitational pull. Objects can be weightless, but they can never be without mass.

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Problem B10

The precision is based on how fine the measuring instrument is. In a two-pan balance, precision is based on the values of the pan weights being used. The smaller the value of the unit, the more precise the measurement. For example, measurements made using milligrams are more precise than those using grams or kilograms.

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Problem B11

The type of measure you use depends on what you're really looking for, since there is no absolute meaning of "largest." In a group of rocks, one may have the greatest surface area, another may have the greatest volume, and a third may have the greatest weight. The meaning of "largest" depends on circumstances and the judgement of those involved in the decision-making process. So, for example, if you decide that the largest rock is the heaviest rock, you would use a scale, rather than the tinfoil-and-grid-paper or water-displacement methods.

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Problem B12

The fault in this line of reasoning is that your rock does not have the same density as water. For most rocks, 1 cm3 of rock weighs more than 1 cm3 of water. This can be seen by noting whether the rock sinks or floats when placed in water. If it sinks, it is denser than water and has more weight than the same volume of water.

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Problem B13

You can choose to measure length in any direction, once the rock is placed in a particular orientation. Or you might choose to measure the circumference of the rock. As with surface area, volume, and weight, none of these linear measurements could be used by itself to determine if a rock is "largest" -- unless that's your only criterion!

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