Difference between revisions of "2013 AIME II Problems/Problem 3"
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A large candle is <math>119</math> centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes <math>10</math> seconds to burn down the first centimeter from the top, <math>20</math> seconds to burn down the second centimeter, and <math>10k</math> seconds to burn down the <math>k</math>-th centimeter. Suppose it takes <math>T</math> seconds for the candle to burn down completely. Then <math>\tfrac{T}{2}</math> seconds after it is lit, the candle's height in centimeters will be <math>h</math>. Find <math>10h</math>. | A large candle is <math>119</math> centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes <math>10</math> seconds to burn down the first centimeter from the top, <math>20</math> seconds to burn down the second centimeter, and <math>10k</math> seconds to burn down the <math>k</math>-th centimeter. Suppose it takes <math>T</math> seconds for the candle to burn down completely. Then <math>\tfrac{T}{2}</math> seconds after it is lit, the candle's height in centimeters will be <math>h</math>. Find <math>10h</math>. | ||
Revision as of 16:50, 6 April 2013
Problem 3
A large candle is centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes
seconds to burn down the first centimeter from the top,
seconds to burn down the second centimeter, and
seconds to burn down the
-th centimeter. Suppose it takes
seconds for the candle to burn down completely. Then
seconds after it is lit, the candle's height in centimeters will be
. Find
.
Solution
We find that . From Gauss' formula, we find that the value of T is
. The value of
is therefore
. We find that
is
, so
. As a result,
, which leads to
. We notice that
, so the answer is
.
See also
2013 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |