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प्रश्न
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 |
उत्तर
Here the intervals are unequal.
By Lagrange’s formula we have,
x0 = 1982
x1 = 1983
x2 = 1984
x3 = 1986
y0 = 150
y1 = 235
y2 = 365
y3 = 525 and x = 1985.
y = `"f"(x) = ((x - x_1)(x - x_2)(x - x_3))/((x_0 - x_1)(x_0 - x_2)(x_0 - x_3)) xx y_0 + ((x - x_0)(x - x_2)(x - x_3))/((x_1 - x_0)(x_1 - x_2)(x_1 - x_3)) xx y_1 + ((x - x_0)(x - x_1)(x - x_3))/((x _2 - x_0)(x_2 - x_1)(x_2 - x_3)) xx y_2 + ((x - x_0)(x - x_1)(x - x_2))/((x_3 - x_0)(x_3 - x_1)(x_3 - x_2)) xx y_3`
y = `((1985 - 1983)(1985 - 1984)(1985 - 1986))/((1982 - 1983)(1982 - 1984)(1982 - 1986)) xx 150 + ((1985 - 1982)(1985 - 1984)(1985 - 1986))/((1983 - 1982)(1983 - 1984)(1983 - 1986)) xx 235 ((198 - 982)(1985 - 1983)(985 - 1986))/((984 - 1982)(1984 - 1983)(1984 - 1986)) xx 365 + ((1985 - 1982)(1985 - 1983)(1985 - 1984))/((1986 - 1982)(1986 - 1983)(1986 - 1984)) xx 525`
= `((2)(1)(-1))/((-1)(-2)(-4)) xx 150 + ((3)(1)(-1))/((1)(-1)(-3)) xx 235 + ((3)(2)(-1))/((2)(1)(-)) xx 365 + ((3)(2)(1))/((4)(3)(2)) xx 525`
= `(-2 xx 150)/(-8) + (-3 xx 235)/3 + (-6 xx 365)/(-4) xx (6 xx 525)/24`
= 37.5 – 235 + 547.5 + 131.25
= 716.125 – 235
= 481.25
∴ Business done in the year 1985 is 481.25 lakhs.
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