1989 AHSME Problems/Problem 28
Contents
Problem
Find the sum of the roots of that are between
and
radians.
Solution
The roots of are positive and distinct, so by considering the graph of
, the smallest two roots of the original equation
are between
and
, and the two other roots are
.
Then from the quadratic equation we discover that the product which implies that
does not exist. The bounds then imply that
. Thus
which is
.
Second Solution
:
We treat
and
as the roots of our equation.
Because
by Vieta's formula,
.
Because the principal values of
and
are acute and our range for
is
,
we have four values of
that satisfy the quadratic:
Summing these, we obtain
.
Using the fact that
,
we get
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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