Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 2"
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==Problem== | ==Problem== | ||
− | Two | + | Two [[point]]s <math>A(x_1, y_1)</math> and <math>B(x_2, y_2)</math> are chosen on the [[graph]] of <math>f(x) = \ln x</math>, with <math>0 < x_1 < x_2</math>. The points <math>C</math> and <math>D</math> trisect <math>\overline{AB}</math>, with <math>AC < CB</math>. Through <math>C</math> a horizontal [[line]] is drawn to cut the curve at <math>E(x_3, y_3)</math>. Find <math>x_3</math> if <math>x_1 = 1</math> and <math>x_2 = 1000</math>. |
==Solution== | ==Solution== | ||
− | Since <math>C</math> is the trisector of [[line segment]] <math>\overline{AB}</math> closer to <math>A</math>, the <math>y</math>-coordinate of <math>C</math> is equal to two thirds the <math>y</math>-coordinate of <math>A</math> plus one third the <math>y</math>-coordinate of <math>B</math>. Thus, point <math>C</math> has coordinates <math>(x_0, \frac{2}{3} \ln 1 + \frac{1}{3}\ln 1000) = (x_0, \ln 10)</math> for some <math> | + | Since <math>C</math> is the trisector of [[line segment]] <math>\overline{AB}</math> closer to <math>A</math>, the <math>y</math>-coordinate of <math>C</math> is equal to two thirds the <math>y</math>-coordinate of <math>A</math> plus one third the <math>y</math>-coordinate of <math>B</math>. Thus, point <math>C</math> has coordinates <math>(x_0, \frac{2}{3} \ln 1 + \frac{1}{3}\ln 1000) = (x_0, \ln 10)</math> for some <math>x_0</math>. Then the horizontal [[line]] through <math>C</math> has equation <math>y = \ln 10</math>, and this intersects the curve <math>y = \ln x</math> at the point <math>(10, \ln 10)</math>, so <math>x_3 = 10</math>. |
+ | ==See also== | ||
+ | {{Mock AIME box|year=2006-2007|n=4|num-b=1|num-a=3|source=125025}} | ||
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*[[Logarithm]] | *[[Logarithm]] | ||
*[[Coordinate geometry]] | *[[Coordinate geometry]] | ||
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+ | [[Category:Intermediate Geometry Problems]] | ||
+ | [[Category:Intermediate Algebra Problems]] |
Latest revision as of 15:25, 8 October 2007
Problem
Two points and
are chosen on the graph of
, with
. The points
and
trisect
, with
. Through
a horizontal line is drawn to cut the curve at
. Find
if
and
.
Solution
Since is the trisector of line segment
closer to
, the
-coordinate of
is equal to two thirds the
-coordinate of
plus one third the
-coordinate of
. Thus, point
has coordinates
for some
. Then the horizontal line through
has equation
, and this intersects the curve
at the point
, so
.
See also
Mock AIME 4 2006-2007 (Problems, Source) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |