STEP II 2007 question 1 solution

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STEP II 2007 question 1 solution

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TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP II 2007 question 1 solution

i) $\left( 1 + \frac{k}{100} \right)^{\frac{1}{2}} \approx 1 + \frac{k}{200} - \frac{k^2}{80000} + \frac{k^3}{16000000}$

a) $\left( 1 + \frac{8}{100} \right)^{\frac{1}{2}} = \left( \frac{27}{25} \right)^{\frac{1}{2}} = \frac{3}{5} \sqrt{3}$

So $\sqrt{3} \approx \frac{5}{3} \left( 1 + \frac{8}{200} - \frac{8^2}{80000} + \frac{8^3}{16000000} \right)$

$= \frac{5}{3} \left( 1 + 0.04 - 0.0008 + 0.000032 \right) = \frac{5}{3} \left( 1.039232 \right)$

$= \frac{10.39232}{6} \approx 1.73205$

By long division.

b) Take k=-4. Then:

$\left( 1 + \frac{-4}{100} \right)^{\frac{1}{2}} = \left( \frac{24}{25} \right)^{\frac{1}{2}} = \frac{2}{5} \sqrt{6}$

$\sqrt{6} \approx \frac{5}{2} \left( 1 - \frac{4}{200} - \frac{4^2}{80000} - \frac{4^3}{16000000} \right)$

$= \frac{5}{2} \left( 1 - 0.02 - 0.0002 - 0.000004 \right) = \frac{5}{2} \left( 0.979796 \right)$

$= \frac{9.79796}{4} \approx 2.44949$

By long division.

ii) $\left( 1 + \frac{k}{1000} \right)^{\frac{1}{3}} \approx \frac{3000 + k}{3000}$

Pick k=29. Then $\left( 1 + \frac{29}{1000} \right)^{\frac{1}{3}} = \left( \frac{1029}{1000} \right)^{\frac{1}{3}} = \left( \frac{3 \times 343}{1000} \right)^{\frac{1}{3}} = \frac{7}{10} \sqrt[3]{3}$