Limits of Accuracy

From: Tom Warren
Date: 21 October 1999

Subject: Limits of Accuracy

A sports commentator is trying to predict the winning margin, t seconds, of racing driver A over his nearest rival B.
He has calculated that A’s average practice speed for the course is 220.8 kmph and B’s average practice speed is 220.4 kmph.
The race is over 50 laps of a 3.84km circuit.

Use this information to calculate t and
the limits within which it might be expected to lie.

Maths Help suggests:

To predict the value of t, we need to use the formula:

Distance = Average Speed × Time

We can work out the times taken by each driver and subtract to find t.

For driver A

This translates into 52 min 10.43 sec

For driver B

An equivalent calculation gives 52 min 16.12 sec

Calculating t

Subtracting times gives t = 5.68 seconds.

Upper and Lower Limits of t

Driver A's speed has been rounded to 220.8 km per hour.
The actual speed could lie between 220.75 and 220.8499999... kmph.
(A speed below 220.75 would be rounded down to 220.7;
A speed of 220.85 or above would be rounded up to 220.9)

In practice, we quote the limits as 220.75 and 220.85 kmph

The table shows the limits of accuracy for the other measurements.

	Speed of A	Speed of B	Length of Circuit
Greatest	220.85	220.45	3.845
Least	220.75	220.35	3.835

Limits for t

The greatest value of t occurs when:

A's speed is at its maximum limit, and
B's speed is at its lowest limit,
over the longest possible circuit length.

This works out to be 7.11 seconds.

The least value of t occurs when:

A's speed is at its lowest, and
B's speed is at its highest,
over the shortest circuit length.

This works out to be 4.22 seconds.

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