Difference between revisions of "2014 AMC 12B Problems/Problem 10"
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<cmath>a^2 + b^2 + c^2 = 36 + 0 + 1 = \boxed{\textbf{(D)}\ 37}</cmath> | <cmath>a^2 + b^2 + c^2 = 36 + 0 + 1 = \boxed{\textbf{(D)}\ 37}</cmath> | ||
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{{AMC12 box|year=2014|ab=B|num-b=9|num-a=11}} | {{AMC12 box|year=2014|ab=B|num-b=9|num-a=11}} | ||
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Revision as of 12:23, 22 February 2014
Problem
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, 
miles was displayed on the odometer, where is a 3-digit number with 
and 
. At the end of the trip, the odometer showed 
miles. What is 
.
$\textbf{(A)}\ 26\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}}\ 37\qquad\textbf{(E)}\ 41$ (Error compiling LaTeX. Unknown error_msg)
Solution
We know that the number of miles she drove is divisible by 
, so 
and 
must either be the equal or differ by . We can quickly conclude that the former is impossible, so and must be apart. Because we know that 
and 
and 
, we find that the only possible values for and are 
and 
, respectively. Because 
, 
. Therefore, we have
![\[a^2 + b^2 + c^2 = 36 + 0 + 1 = \boxed{\textbf{(D)}\ 37}\]](http://latex.artofproblemsolving.com/1/a/3/1a3c061138770801170b4377f6deae8a9fb4a2a3.png)
See also
| 2014 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
