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Error | IB Math AIHL | Perplex

Approximations & Error (Lesson 3/3)

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Discussion

We saw in the rounding numbers section that the level of precision available when measuring values can impact the final answer of a calculation. Finding the volume of water in a graduated cylinder with markings every 1 mL, for example, could result in a different answer than finding the volume of the same amount of water in a different graduated cylinder that has markings every 0.25 mL.

Both of these cylinders have the same amount of water in them, but if we took the measurement using the first cylinder we'd get 25 mL, and if we took it using the second, we'd get 25.25 mL.

What range of numbers do you think the "actual" volume of the water might fall in, based off of the measurement given by first cylinder? Based off of the second?

Solution:

When you read 25mL on a scale marked every 1mL, the water level must be closer to the 25mL mark than to either the 24mL or the 26mL marks, but you do not know exactly where between them. The point where it would instead look closer to 24mL is halfway between 24 and 25, which is

224+25=24.5

and the point where it would look closer to 26mL is halfway between 25 and 26, which is

225+26=25.5

So any true volume between 24.5mL and 25.5mL would still be read as 25mL. Therefore,

V∈[24.5,25.5]mL

On the 0.25mL-graduation cylinder, a reading of 25.25mL means the water level is nearer the 25.25 mark than to 25.00 or 25.50. The halfway points are

225.00+25.25=25.125and225.25+25.50=25.375

So any true volume between 25.125mL and 25.375mL would register as 25.25mL, giving

V∈[25.125,25.375]mL

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