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1 year ago

Approximate pi plus the square root of 70 to the nearest hundredth.

Mathematics

1 answer:

stich3 [128]1 year ago

5 0

Answer:

11.51

Step-by-step explanation:

In general, a calculator is required to evaluate the irrational sum ...

π + √70

<h3>Calculator result</h3>

The result from a readily-available online calculator is shown in the attachment.

π + √70 ≈ 11.51

<em>Additional comment</em>

You have probably memorized pi to several digits: 3.1416. To obtain the required estimate means you need to approximate √70 to about the same level of precision.

For a number n = a² +b, the square root can be approximated by the continued fraction ...

√n ≈ a +b/(2a +b/(2a +...))

Effectively, the root (r) can be iterated from the recursive formula ...

r' = a +b/(a +r)

For √70, we have 70 = 8² +6 ⇒ a=8, b=6

If the initial approximation of the root is r0≈a=8, then a couple of iterations gives ...

r1 = 8 + 6/(8 +r0) = 8 + 6/(8 +8) = 8 3/8

r2 = 8 + 6/(8 +r1) = 8 +6/(8 +8 3/8) = 8 48/131

This has sufficient accuracy for the purpose. The decimal equivalent of the sum then becomes ...

3.1416 +8.3664 = 11.5080 ≈ 11.51

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