Dear Class 12 Samacheer Kalvi students, here are the Business Maths Chapter 5 Numerical Methods Exercise 5.2 Text Book Solutions for your study.
Numerical Methods Exercise 5.2 Text Book Solutions
1. Using graphic method, find the value of y when x = 48 from the following data:
x | 40 | 50 | 60 | 70 |
y | 6.2 | 7.2 | 9.1 | 12 |
Answer: Plot the points (40,6.2), (50,7.2), (60,9.1) and (70,12)
Scale: x axis: 1 cm = 10 units; y axis: 1 cm = 1 unit;
2. The following data relates to indirect labour expenses and the level of output.
Months | Jan | Feb | Mar | Apr | May | Jun |
Units of output | 200 | 300 | 400 | 640 | 540 | 580 |
Indirect labour expenses (Rs.) | 2500 | 2800 | 3100 | 3820 | 3220 | 3640 |
Estimate the expenses at a level of output of 350 units, by using graphic method.
Answer: Plot the points (200,2500), (300,2800), (400,3100), (640, 3820), (540, 3220), (580,3640)
From the graph, when x=350, y=2900.
Expenses of 350 units = Rs. 2900
Scale: x axis: 1 cm = 100 units; y axis: 1 cm = 500 units
3. Using Newton’s forward interpolation formula find the cubic polynomial.
x | 0 | 1 | 2 | 3 |
f(x) | 1 | 2 | 1 | 10 |
Answer: Newton’s forward interpolation formula
The difference table is:
x | y | ∆y | ∆2 y | ∆3 y |
0 | 1 | |||
1 | 2 | 1 | ||
2 | 1 | -1 | -2 | |
3 | 10 | 9 | 10 | 12 |
4. The population of a city in a censes taken once in 10 years is given below. Estimate the population in the year 1955.
Year | 1951 | 1961 | 1971 | 1981 |
Population in lakhs | 35 | 42 | 58 | 84 |
Since 1955 lies between 1951 and 1961, we can use Newton’s forward interpolation formula.
The difference table is:
x | y | ∆y | ∆2 y | ∆3 y |
1951 | 35 | |||
1961 | 42 | 7 | ||
1971 | 58 | 16 | 9 | |
1981 | 84 | 26 | 10 | 1 |
5. In an examination the number of candidates who secured marks between certain interval were as follows:
Marks | 0-19 | 20-39 | 40-59 | 60-79 | 80-99 |
No of candidates | 41 | 62 | 65 | 50 | 17 |
Estimate the number of candidates whose marks are less than 70.
Marks | No of candidates | Cumulative frequency |
-.5-19.5 | 41 | 41 |
19.5-39.5 | 62 | 103 |
39.5-59.5 | 65 | 168 |
59.5-79.5 | 50 | 218 |
79.5-99.5 | 17 | 235 |
x | y | ∇y | ∇2 y | ∇3 y | ∇4 y |
Less than 19.5 | 41 | ||||
Less than 39.5 | 103 | 62 | |||
Less than 59.5 | 168 | 65 | 3 | ||
Less than 79.5 | 218 | 50 | -15 | -18 | |
Less than 99.5 | 235 | 17 | -33 | -18 | 0 |
Since we need the number of candidates whose marks are less than 70, we can use Newton’s backward interpolation formula.
6. Find the value of f (x) when x = 32 from the following table:
x | 30 | 35 | 40 | 45 | 50 |
f(x) | 15.9 | 14.9 | 14.1 | 13.3 | 12.5 |
Since x=32 lies between 30 and 35, we can use Newton’s forward interpolation formula.
x | y | ∆y | ∆2 y | ∆3 y | ∆4 y |
30 | 15.9 | ||||
35 | 14.9 | -1 | |||
40 | 14.1 | -0.8 | 0.2 | ||
45 | 13.3 | -0.8 | 0 | -0.2 | |
50 | 12.5 | -0.8 | 0 | 0 | 0.2 |
7. The following data gives the melting point of a alloy of lead and zinc where ‘t’ is the temperature in degree c and P is the percentage of lead in the alloy.
P | 40 | 50 | 60 | 70 | 80 | 90 |
T | 180 | 204 | 226 | 250 | 276 | 304 |
Find the melting point of the alloy containing 84 percent lead.
Since x=84 lies between 80 and 90, we can use Newton’s backward interpolation formula.
x | y | ∇y | ∇2 y | ∇3 y | ∇4 y | ∇4 y |
40 | 180 | |||||
50 | 204 | 24 | ||||
60 | 226 | 22 | -2 | |||
70 | 250 | 24 | 2 | 4 | ||
80 | 276 | 26 | 2 | 0 | -4 | |
90 | 304 | 28 | 2 | 0 | 0 | 4 |
8. Find f (2.8) from the following table:
x | 0 | 1 | 2 | 3 |
f(x) | 1 | 2 | 11 | 34 |
Since 2.8 lies between 2 and 3, we can use Newton’s backward interpolation formula.
x | y | ∇y | ∇2 y | ∇3 y |
0 | 1 | |||
1 | 2 | 1 | ||
2 | 11 | 9 | 8 | |
3 | 34 | 23 | 14 | 6 |
9. Using interpolation estimate the output of a factory in 1986 from the following data.
Year | 1974 | 1978 | 1982 | 1990 |
Output in 1000 tonnes | 25 | 60 | 80 | 170 |
Answer: Since the intervals are unequal, we can use LaGrange’s formula.
10. Use Lagrange’s formula and estimate from the following data the number of workers getting income not exceeding Rs. 26 per month.
Income not exceeding(Rs.) | 15 | 25 | 30 | 35 |
No. of workers | 36 | 40 | 45 | 48 |
11. Using interpolation estimate the business done in 1985 from the following data.
Year | 1982 | 1983 | 1984 | 1986 |
Business done (in lakhs) | 150 | 235 | 365 | 525 |
Answer: Since the intervals are unequal, we can use LaGrange’s formula.
12. Using interpolation, find the value of f(x) when x = 15
x | 3 | 7 | 11 | 19 |
f(x) | 42 | 43 | 47 | 60 |
Answer: Since the intervals are unequal, we can use LaGrange’s formula.
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