Difference between revisions of "2004 AMC 12B Problems/Problem 6"
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== Problem == | == 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? | + | Minneapolis-St. Paul International Airport is <math>8</math> miles southwest of downtown St. Paul and <math>10</math> miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? |
| − | <math> | + | <math>\mathrm{(A)\ }13\qquad\mathrm{(B)\ }14\qquad\mathrm{(C)\ }15\qquad\mathrm{(D)\ }16\qquad\mathrm{(E)\ }17</math> |
== Solution == | == 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 <math>\sqrt{8^2 + 10^2} \ | + | 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>. |
| − | Without a calculator one can note that <math>8^2 + 10^2 = 164 < 169 = 13^2</math>. | + | 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 == | ||
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{{AMC12 box|year=2004|ab=B|num-b=5|num-a=7}} | {{AMC12 box|year=2004|ab=B|num-b=5|num-a=7}} | ||
{{AMC10 box|year=2004|ab=B|num-b=7|num-a=9}} | {{AMC10 box|year=2004|ab=B|num-b=7|num-a=9}} | ||
| + | {{MAA Notice}} | ||
Revision as of 21:18, 12 June 2015
- 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 
miles southwest of downtown St. Paul and 
miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
Solution
The directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs miles and miles long. The hypotenuse length is 
, and thus the answer is 
.
Without a calculator one can note that 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
