Question

The following are the data on the drying time of a certain varnish and the amount of an additive that is intended to reduce the drying time? Amount of varnish additive(grams)""x"" 0 1 2 3 4 5 6 7 8 Drying time(hr) ""y"" 12.0 10.5 10.0 8.0 7.0 8.0 7.5 8.5 9.0 (i) Fit a second degree polynomial by the method of least square. (ii) Use the result of (i) to predict the drying time of the varnish when 6.5 gms of the additive is being used.

Answer 1

Answer:

The equation is y=a+bx+cx2 and the normal equations are

∑y=an+b∑x+c∑x2

∑xy=a∑x+b∑x2+c∑x3

∑x2y=a∑x2+b∑x3+c∑x4

The values are calculated using the following table

x y x2 x3 x4 x⋅y x2⋅y

0 12 0 0 0 0 0

1 10.5 1 1 1 10.5 10.5

2 10 4 8 16  20 40

3 8 9 27 81  24 72

4 7 16 64 256  28 112

5 8 25 125 625  40 200

6 7.5 36 216 1296  45 270

7 8.5 49 343 2401  59.5 416.5

8 9 64 512 4096  72 576

--- --- --- --- --- --- ---

∑x=36 ∑y=80.5 ∑x2=204 ∑x3=1296 ∑x4=8772 ∑x⋅y=299 ∑x2⋅y=1697

Substituting these values in the normal equations

9a+36b+204c=80.5

36a+204b+1296c=299

204a+1296b+8772c=1697

Solving these 3 equations,

Total Equations are 3

9a+36b+204c=80.5→(1)

36a+204b+1296c=299→(2)

204a+1296b+8772c=1697→(3)

Select the equations (1) and (2), and eliminate the variable a.

9a+36b+204c=80.5 ×4→   36a + 144b + 816c = 322  

−  

36a+204b+1296c=299 ×1→   36a + 204b + 1296c = 299  

- 60b - 480c = 23 →(4)

Select the equations (1) and (3), and eliminate the variable a.

9a+36b+204c=80.5 ×68→   612a + 2448b + 13872c = 5474  

−  

204a+1296b+8772c=1697 ×3→   612a + 3888b + 26316c = 5091  

- 1440b - 12444c = 383 →(5)

Answer 2

Step-by-step explanation: