Unit 1: Limits to Measurement - WordPress.com...Math 3202 Section 1.1 Limits to Measurement Notes 1 Unit 1: Limits to Measurement 0.001 0.01 0.1 1 10 100 1000 Activate Prior Knowledge:

Math 3202 Section 1.1 Limits to Measurement Notes

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Unit 1: Limits to Measurement

1 10 100 10000.10.010.001

Activate Prior Knowledge: Metric Conversions

In the Metric System, the prefixes tell you the relationship between the numbers. For measurements of length, the base unit is the metre (m). The following lengths are equivalent.

One metre is equal to 1000 mm or 0.001 km

One kilometre is equal to 1000 m.

Example

65 km = _____________ m

1200 m = ____________ km

400 cm = ____________ m

5600 mm = __________ cm

98 000 000 000 mm = _____ km

Jonathan Mauger

Text Box

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Activate Prior Knowledge: RoundingIt is often necessary to round measured values, to show the level of accuracy of your measurements and calculations

Example: Round the following to the nearest tenth.

87.253 Look at the number following the tenths position. If it is 5 or

larger, then the 2 moves up to 3. If it is smaller than 5, the 2 remains the same.

87.253 rounds to 87.3

87.223 rounds to 87.2


Round each measurement to the indicated degree of precision.

3564 m to the nearest 10

42.3979 cm to the nearest tenth

42.3979 cm to the nearest hundredth

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Activate Prior Knowledge: Adding and Subtracting Fractions Remember You NEED COMMON DENOMINATORS to add and subtract fractions.

Change Mixed Fractions to Improper Fractions

Change Improper Fractions to Mixed Fractions

Use Calculator!


Perform the Indicated operations

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Sec 1.1 Accuracy and Precision

Accuracy of a measuring device means how close is the measured value to its true value.

For example: if you use a one cup measure to measure sugar, how close is it to actually being one cup of sugar.

Can other people repeat the measurement with the same or a different measuring cup and get the same result?

Accuracy also depends on how carefully you use the device.

The precision of a measurement is restricted by the limitations of the measurement scale. If the smallest division on your ruler is a millimetre, you can only make a measurement to the nearest millimetre.

Because the precision of a measurement depends on the limitations of the measuring device, there is some uncertainty in a measurement.

Precision of this ruler is 1 millimetreUncertainty is half the measure of precision or ±0.5 mm

measured value ± measurement uncertainty


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A gardener mixes fertilizer with 1 L of water. The water jug has lines showing the volume every 0.1 L. The uncertainty is half the smallest measurement, or 0.05 L. The amount of water in the jug can be as much as 1.05 L or as little as 0.95 L.

measured value = 1.0 ± 0.05 L

If you added another volume of 1.0 ± 0.05 L of fertilizer to the 1.0 ± 0.05 L of fertilizer, what will be the resulting volume?

The uncertainty of the final amount is the sum of the measured uncertainties.

final volume = (1.0 ± 0.05 L) + (1.0 ± 0.05 L)

final volume of fertilizer = 2.0

± 0.1 L

Example 1State the precision and uncertainty of each measurement

a) A baker sets an oven timer that displays 1minute increments to 45 min.

b) A nurse measures a patient's temperature to be

c) A jeweller weighs 1.243 g of gold.

d) A survey technician records the distance between two survey points to be 1200 m.


36.8oC

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Discuss the Ideas: What Level of Precision is Needed?

George is a finishing carpenter in Fort Simpson, NT. He is installing a run of crown moulding along the top of a set of kitchen cupboards.

He has three tape measures, each marked with a different scale. The first tape is marked in centimetres, the second in halfcentimetres, and the third in tenths of a centimetre (millimetres). He uses each tape to measure a piece of moulding.

1. Using Tape #1, George could say that the piece is 12 cm long. The tape doesnot allow him to state a more precise length without estimating a measurement between the marks. Thus, the length is greater than or equal to 11.5 cm and less than or equal to 12.5 cm. This can be written as 12 cm ± 0.5 cm.

a) Write a similar statement describing the measurement for tape #2.

b) Write a similar statement describing the measurement for Tape #3.

c) Which measurement is most precise? Explain your thinking.


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2. It is important that the join between the crown moulding and the cabinets isnot visible. Which level of precision should George choose for his installation? Explain your choice.

3. If George were building a fence, which of the three measuring tapes would

be the best choice for measuring the height of the posts? Explain your choice.

Example 2In winter sports such as alpine skiing, snowboarding and speed skating, the precision and accuracy of the timing devices used are very important. In the downhill alpine ski race for visually impaired women at the 2010 Winter Paralymipic Games in Whistler, BC, the difference in recorded times between firstplace finisher Canadian Viviane Forest (1:27:51) and the secondplace finisher, Henrieta Farkasova of Slovakia, was 0.66 s.

a) What was the precision of the timer used? What was the uncertainty?

b) Write the firstplace finisher's time, including the measurement uncertainty.

c) What was the time of the secondplace finisher, including themeasurement uncertainty? Given the uncertainty, write the maximum and miminum times it could have taken her to finish the race.


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Example 3Carry out the calculations each person is performing. State the answer to the correct level of precision and calculate the uncertainty.

a) A kitchen cabinet installer is laying out a kitchen design. Sheplans to install the fridge and the stove side by side. She measures the fridge to be 71 ± 0.5 cm wide and the stove to be 76 ± 0.5 cm wide. She calculates their combined width.

b) A shipping clerk calculates the total shipping weight of threepackages with individual weights of 16.5 kg, 2.8 kg, and 1.4 kg.


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c) A pipefitter calculates the length of a 2.05m pipe after he has cut a6cm piece from the end.


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Airlines set limits on the weight of luggage that passengers can bring onboard. The limit for a carryon bag is 10 kg. Marc packed some items into his backpack and weighed it on a bathroom scale that measures to a level of precision of 0.1 kg. It weights 3.5 kg. Can he safely add all the

remaining items below, bearing in mind the uncertainty? Explain. How can he reduce the uncertainty in the total weight measurement?

computer 2.5 kg

textbook 1.9 kg

water bottle 1.5 kgnovel 0.5 kg


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1. Carry out the calculations each person is performing and give the correctuncertainty.

a) A dump truck operator estimates the distance (gate to gate) from a gravel pit toa job site to be 9.3 km. He adds 0.5 km for travel inside the job site and 0.3 km at the gravel pit.

b) JeanMarie calculates the clearance between a refrigerator door in its openposition (27 ") and the countertop across from the fridge (32" away). He measures to the nearest quarter of an inch.

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2. Express each measurement in the following form: Measurement ± Uncertainty

a) Ben, a set builder working for Theatre la Seizieme, measures a 2metre longpiece of particle board using a tape measure that is marked in 1mm increments.

b) Sean, a dairy farmer, prepares 15 ml of antibiotic in a syringe marked in 1mlincrements.

c) The speedometer on a car reads 35 km/h with speed marked every 5 km/h.

d) A digital roadside speed indicator indicates 34 km/h.

e) Pia, a sheet metal worker, measures the length (33.5 cm) and width (18.5 cm) ofa piece of sheet metal using a ruler (smallest gradation 1mm)


Assignment: Section 1.1

Name: ______________

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3. a) Leo, a post office employee, calculates the total shipping weight of two packages being sent to Sachs Harbour, NT. The packages weigh 214.3 kg and 5.0 kg. What are the uncertainties of the package weights? What is the combined weight and uncertainty?

b) Leo is packing another box for shipment. He packs in several items and weighsthe box to be 12.8 kg. He realizes the box won't close properly, so he takes out an item weighting 3.4 kg. What are the uncertainties of the package weights? What is the weight of the box after the smaller package has been removed, including the uncertainty?

4. In Sakatoon, Sk, Julia works as a mobile crane operator. She iscalculating the total mass of a load consisting of four pieces weighing 563 kg, 69 kg, 1125 kg, and 10 kg. What are the uncertainties of the individual pieces? What is the total mass of the load and the combined uncertainty?


a) 10.1 km ± 0.15 km

b) 4 " ± "34

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1.

Answers:

2. a) 2000 mm ± 0.5 mmb) 15 ml ± 0.5 mlc) 35 km/h ± 2.5 km/hd) 34 km/h ± 0.5 km/he) length = 33.5 cm ± 0.05 cm

width = 18.5 cm ± 0.05 cm

3. a) The uncertainty of each package weight is ± 0.05 kg219.3 ± 0.1 kg

b) The uncertainty of each package weight is ± 0.05 kg9.4 ± 0.1 kg

4. Each piece has an uncertainty of 0.5 kg

1767 ± 2 kg


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The nominal value is the target value for the measured quantity. It is usually halfway between the maximum and minimum values, but it may also be a minimum or maximum value. The nominal value indicated the value to be aimed for, but an amount of difference above and below this value is acceptable.

The manufacturing tolerance is the difference between the maximum acceptable dimension and the minimum acceptable dimension. It can be expressed in several ways.

*

* nominal value ± (tolerance)

* minimum value

* maximum value

Tolerance in not an uncertainty, but rather the acceptable range of a given measurement

minimum valuemaximum value

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+ tolerance0

tolerance+0


Sec 1.1 Tolerances

You should be as accurate as possible while measuring and make your measurements precise enough for the task you are doing. That is, when you are manufacturing goods, a range of measurements is often acceptable. The range of acceptable dimensions is call the tolerance.

The concept of tolerance is important because when you install and assemble parts, they must fit together, work properly, and perhaps look right.

* Two parts of a hinge must fit together and still slide against each other, but not allowthe door to tilt.

* A kitchen cabinet installation must leave room for the kitchen door to open withoutobstruction.

* A crushed ore screen at a mine must allow only pieces smaller than a particular sizeto pass through.

Small variations occur in manufacturing processes. If a range of

measurements is acceptable, then a higher number of items can be used.

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b) Another dimension on the drawing is specified as.Express this dimension in the following form.nominal value ± 1/2 (tolerance)

0.250"0.230"


Example 1

Lana works as a mechanical designer in Edmonton, ABa) She specifies a dimension on a machining drawing that has a tolerance of

0.020" and a nominal value of 0.500". If the nominal value is the midpoint of the tolerance range, how can the designer write the dimension so the machinist knows the acceptable range of measurements?

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b) In this case, would it be better to make the hole smaller or larger?

c) How might the answer to b) be reflected in the way the tolerance is expressed?

Example 2

Andrij runs a home renovation business. His current job is installing drywall in a home. He needs to cut openings in the drywall for the electrical outlets. Each hole in the drywall must be big enough to clear the electrical box, but small enough that the outlet cover plate will completely cover the edges of the drywall so that it looks tidy.

a) What is the size and tolerance of the hole that Andrij should cut?

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Assignment: Section 1.1

Name: _________________

1. Jim is employed by an automotive repair shop. He is replacing the spark plugs on acar that he is servicing. The manufacturer's specifications say that the spark plug gap (width between the electrodes of the spark plug) should be 0.048" 0.052". What are the tolerance, nominal value, and ± range of the spark plug gap? Assume that the nominal value is the midpoint of the specification range?


2. Carlos works for a machine shop in Portage la Prairie, MB. He is inspectingmachined parts for quality control. The dimension that he is checking is labelled on the drawing as 0.825" ± 0.002". He randomly checks ten parts from three production lines and measures the dimensions show in the table.

a) Determine the maximum and minimum acceptable values of the dimensions.

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b) Complete the table and write down whether each part would be accepted or rejected based on the dimensions given.

c) Based on the inspection data Carlos compiles, what recommendations for actionmight he make?

5.


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