2009 AMC 12A Problems/Problem 13
Problem
A ship sails miles in a straight line from
to
, turns through an angle between
and
, and then sails another
miles to
. Let
be measured in miles. Which of the following intervals contains
?
Solution
Let be the point the ship would reach if it turned
, and
the point it would reach if it turned
. Obviously,
is the furthest possible point from
, and
is the closest possible point to
.
Hence the interval of possible values for is
.
We can find and
as follows:
Let and
be the feet of the heights from
and
onto
. The angles in the triangle
are
,
, and
, hence
. Similarly, the angles in the triangle
are
,
, and
, hence
and
.
Hence we get:
Therefore for any valid the value
is surely in the interval
.
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 |