Question

Calculate the percentage error made in Stirling s formula logn nlogn! = nlogn - n when ( i ) n=4 and n=8​

Answer 1

Answer:

Correct option is

B

24 m/s

Let the total distance be 2x.

For first half distance :

Speed is 20 m/s

Thus time taken t

1

=

20

x

For second half distance :

Speed is 30 m/s

Thus time taken t

2

=

30

x

Average velocity V

avg

=

t

1

+t

2

2x

⟹ V

avg

=

20

x

+

30

x

2x

=

20+30

2×20×30

=24 m/s

Answer 2

Answer:

(i) n = 4 is 1%

(ii) n = 8 is 1%

Step-by-step explanation:

Given:- n = 4 and n = 8

To Find:- percentage error using Stirling's formula.

Solution:-

As we know, Stirling formula n! ≈ $\frac{n^{n} }{e^{n} } \sqrt{2\pi n}$.

(i) For n = 4, n! = 4! = 24

$\frac{n^{n} }{e^{n} } \sqrt{2\pi n}$ = $\frac{4^{4} }{e^{4} } \sqrt{2\pi 4}$

             = $\frac{256}{54.6} \sqrt{25.12}$

             = 23.5

∴ Percentage error = $\frac{24}{23.5}$ = 1%.

(ii) For n=8, n! = 8! = 40320

$\frac{n^{n} }{e^{n} } \sqrt{2\pi n}$ = $\frac{8^{8} }{e^{8} } \sqrt{2\pi 8}$

              = $\frac{16777216 }{2980.95} \sqrt{50.24}$

              = 39902.5

∴ Percentage error = $\frac{40320}{39902.5}$ = 1%

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