Difference between revisions of "1987 AHSME Problems/Problem 19"
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== Solution == | == Solution == | ||
− | We have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - | + | We have <math>\sqrt{65} > 8 > 7.5</math>. Also <math>7.5^2 = (7 + 0.5)^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2 = 49 + 7 + 0.25 = 56.25 < 63</math>, so <math>\sqrt{63} > 7.5</math>. Thus <math>\sqrt{65} + \sqrt{63} > 7.5 + 7.5 = 15</math>. Now notice that <math>\sqrt{65} - \sqrt{63} = \frac{(\sqrt{65} - \sqrt{63})(\sqrt{65} + \sqrt{63})}{\sqrt{65} + \sqrt{63}} = \frac{2}{\sqrt{65} + \sqrt{63}}</math>, so <math>\sqrt{65} - \sqrt{63} < \frac{2}{15} = 0.1333333...</math>, so the answer must be <math>A</math> or <math>B</math>. To determine which, we write <math>\sqrt{65} - \sqrt{63} > 0.125 \iff 65 - 2\sqrt{65 \cdot 63} + 63 > 0.015625 \iff 128 - 0.015625 > 2\sqrt{4095} \iff \sqrt{4095} < 64 - 0.0078125 \iff 4095 < 4096 - 128 \cdot 0.0078125 + 0.0078125^2 = 4096 - 1 + 0.0078125^2</math> which is true. Hence as the expression is greater than <math>0.125</math>, and less than or equal to <math>0.13</math> (since we showed it is certainly less than <math>0.1333333...</math>), it is closest to <math>0.13</math>, which is answer <math>\boxed{\text{B}}</math>. |
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+ | (<math>\sqrt{65} - \sqrt{63}</math> is approximately equal to <math>0.125003815</math>) | ||
== See also == | == See also == |
Latest revision as of 07:02, 12 July 2018
Problem
Which of the following is closest to ?
Solution
We have . Also , so . Thus . Now notice that , so , so the answer must be or . To determine which, we write which is true. Hence as the expression is greater than , and less than or equal to (since we showed it is certainly less than ), it is closest to , which is answer .
( is approximately equal to )
See also
1987 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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