basic_engineering_mathematics0

22 Basic Engineering Mathematics
(a) 72 × 31.429 = 2262.888 (by calculator), hence a
rounding-off error has occurred. The answer should
have stated:
72 × 31.429 = 2262.9, correct to 5 significant figures or
2262.9, correct to 1 decimal place.
(b) 16 × 0.08 × 7 = 4 ✚16 × 8
✟100 × 7
25
= 32 × 7
25
= 8.96
= 224
25 = 8 24
25
Hence an order of magnitude error has occurred.
(c) 11.714 × 0.0088 is approximately equal to 12 × 9 × 10 −3 ,
i.e. about 108 × 10 −3 or 0.108. Thus a blunder has been
made.
(d)
(a)
29.74 × 0.0512
11.89
≈
30 × 5 × 10−2
12
= 15
120 = 1 8
= 150
12 × 10 2
or 0.125
hence no order of magnitude error has occurred.
29.74 × 0.0512
However,
= 0.128 correct to 3 significant
figures, which equals 0.13 correct to 2 significant
11.89
figures.
Hence a rounding-off error has occurred.
Problem 3. Without using a calculator, determine an
approximate value of
(a)
11.7 × 19.1
9.3 × 5.7
11.7 × 19.1
9.3 × 5.7
(By calculator,
figures.)
(b)
2.19 × 203.6 × 17.91
(b) ≈
12.1 × 8.76
cancelling
2.19 × 203.6 × 17.91
12.1 × 8.76
10 × 20
is approximately equal to , i.e. about 4
10 × 5
11.7 × 19.1
= 4.22, correct to 3 significant
9.3 × 5.7
2 × 200 × 20
10 × 10
2.19 × 203.6 × 17.91
i.e. ≈ 80
12.1 × 8.76
2.19 × 203.6 × 17.91
(By calculator,
12.1 × 8.76
3 significant figures.)
= 2 × 2 × 20 after
= 75.3, correct to
Now try the following exercise
Exercise 13 Further problems on errors (Answers
on page 271)
In Problems 1 to 5 state which type of error, or errors, have
been made:
1. 25 × 0.06 × 1.4 = 0.21
2. 137 × 6.842 = 937.4
3.
24 × 0.008
12.6
= 10.42
4. For a gas pV = c. When pressure p = 1 03 400 Pa and
V = 0.54 m 3 then c = 55 836 Pa m 3 .
5.
4.6 × 0.07
52.3 × 0.274 = 0.225
In Problems 6 to 8, evaluate the expressions approximately,
without using a calculator.
6. 4.7 × 6.3
7.
8.
2.87 × 4.07
6.12 × 0.96
72.1 × 1.96 × 48.6
139.3 × 5.2
4.2 Use of calculator
The most modern aid to calculations is the pocket-sized electronic
calculator. With one of these, calculations can be quickly
and accurately performed, correct to about 9 significant figures.
The scientific type of calculator has made the use of tables and
logarithms largely redundant.
To help you to become competent at using your calculator
check that you agree with the answers to the following problems:
Problem 4.
figures:
Evaluate the following, correct to 4 significant
(a) 4.7826 + 0.02713 (b) 17.6941 − 11.8762
(c) 21.93 × 0.012981
(a) 4.7826 + 0.02713 = 4.80973 = 4.810, correct to 4 significant
figures.
(b) 17.6941 − 11.8762 = 5.8179 = 5.818, correct to 4 significant
figures.
(c) 21.93 × 0.012981 = 0.2846733 ...= 0.2847, correct to
4 significant figures.