Measurement, Rounding Error, and Reasonableness

There is no such thing as an exact measurement. Measurements are always approximate. No matter how precise it is, a measurement actually represents a range of values.

Example 1

Chris's height, to the nearest inch, is 5 ft 8 in. What range of values does this measurement represent?

The height is given to the nearest inch, so the error is 1 2 1 half  in. Chris's height, then, is between 5 ft 7 1 2 7 , and 1 half  in. and 5 ft 8 1 2 8 , and 1 half  in., or 5 ft 8 in. ± 1 2 plus minus , 1 half  in. Within this range are all the measures that, when rounded to the nearest inch, equal 5 ft 8 in.

As you calculate with measurements, errors can accumulate.

Example 2

Jean drives 18 km to work each day. This distance is given to the nearest kilometer. What is the range of values for the round-trip distance?

The driving distance is between 17.5 and 18.5 km, or 18 ± 0.5  km . 18 plus minus 0.5 , km .  Double the lower limit, 17.5, and the upper limit, 18.5. Thus, the round trip can be anywhere between 35 and 37 km, or 36 ± 1  km . 36 plus minus 1 km .  Notice that the error for the round trip is double the error for a single leg of the trip.

So that your answers will be reasonable, keep precision and error in mind as you calculate. For example, in finding AB, the length of the hypotenuse of Δ A B C , cap delta eh b c comma  it would be inappropriate to give the answer as 8.6533 if the sides are given to the nearest tenth. Round your answer to 8.7.

Exercises

Each measurement is followed by its unit of greatest precision. Find the range of values that each measurement represents.

1. 24 ft (ft)
2. 124 cm (cm)
3. 340 mL (mL)
4. 5 1 2 mi . ( 1 2 mi ) 5 , and 1 half , mi . open , 1 half , mi close
5. 73.2 mm (0.1 mm)
6. 34  yd 2 ( yd 2 ) 34 , yd squared , open , yd squared , close

7. The lengths of the sides of TJCM are given to the nearest tenth of a centimeter. What is the range of values for the figure's perimeter?

   

8. To the nearest degree, two angles of a triangle are 49 ° 49 degrees  and 73 ° . 73 degrees .  What is the range of values for the measure of the third angle?
9. The lengths of the legs of a right triangle are measured as 131 m and 162 m. You use a calculator to find the length of the hypotenuse. The calculator display reads 208.33867. What should your answer be?

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