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Difference between revisions of "2004 AMC 12B Problems/Problem 6"

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== Solution ==
 
== Solution ==
  
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The directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs 8 miles and 10 miles long. The hypotenuse length is <math>\sqrt{8^2 + 10^2} \sim 12.8</math>, and thus the answer is <math>\mathrm{(A)}</math>.
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The directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs <math>8</math> miles and <math>10</math> miles long. The hypotenuse length is <math>\sqrt{8^2 + 10^2}\approx12.8</math>, and thus the answer is <math>\boxed{\mathrm{(A)}\ 13}</math>.
  
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Without a calculator one can note that <math>8^2 + 10^2 = 164 < 169 = 13^2\rightarrow\boxed{A}</math>.
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Without a calculator one can note that <math>8^2+10^2=164<169=13^2\Rightarrow\boxed{\mathrm{(A)}\ 13}</math>.
  
 
== See Also ==
 
== See Also ==

Revision as of 20:31, 22 July 2014

The following problem is from both the 2004 AMC 12B #6 and 2004 AMC 10B #8, so both problems redirect to this page.

Problem

Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?

$\mathrm{(A)\ }13\qquad\mathrm{(B)\ }14\qquad\mathrm{(C)\ }15\qquad\mathrm{(D)\ }16\qquad\mathrm{(E)\ }17$

Solution

The directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs $8$ miles and $10$ miles long. The hypotenuse length is $\sqrt{8^2 + 10^2}\approx12.8$, and thus the answer is $\boxed{\mathrm{(A)}\ 13}$.

Without a calculator one can note that $8^2+10^2=164<169=13^2\Rightarrow\boxed{\mathrm{(A)}\ 13}$.

See Also

2004 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions
2004 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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