Question

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.

Answer 1

Since the required value is at the end of the table

Apply backward interpolation formula.

To find T at p = 84

`"T"_(("p" = "p"_0 + "nh")) = "T"_"n" + "n"/(1!) ∇"T"_"n" + ("n"("n" + 1))/(2!) ∇^2"T"_0 + ("n"("n" + 1)("n"  2))/(3!) Delta^3"T"_0 + .......`

To find T at P = 84

P_n + nh = 84

90 + n(10) = 84

10n = 84 – 90

10n = – 6

⇒ n = `(-6)/10`

n = – 0.6

P	T	`Delta"T"`	`Delta^2"T"`	`Delta^3"T"`	`Delta^4"T"`	`Delta^5"T"`
40	180
		24
50	204		– 2
		22		4
60	226		2		– 4
		24		0		4
70	250		2		0
		26		0
80	276		2
		28
90	304

`"T"_((84)) = 304 + ((-0.6))/(1!) (28) + ((-0.6)(-0.6 + 1))/(2!) (2)+ ((-0.6)(-0.6 + 1)(-0.6 + 2))/(3!) (0) + ((-0.6)(-0.6 + 1)(-0.6 + 2)(0.6 + 3))/(4!) (0) + ((-0.6)(-0.6 + 1)(-0.6 +2)(-0.6 + 3)(-0.6 + 4))/(5!) (4) +`

= 304 – 16.8 – 0.24 – 0.091392

= 304 – 17.131392

= 286.86

Hence the melting point of the alloy is 286.86°c.