Difference between revisions of "2009 AMC 12B Problems/Problem 15"
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<math>\textbf{(D)}\ 3(1 + \sqrt {r})^x = 7\qquad \textbf{(E)}\ 3(1 + 1/r)^x = 7</math> | <math>\textbf{(D)}\ 3(1 + \sqrt {r})^x = 7\qquad \textbf{(E)}\ 3(1 + 1/r)^x = 7</math> | ||
− | == Solution == | + | == Solution 1 == |
'''(B)''' Intuitively, <math>x</math> will be largest for that option for which the value in the parentheses is smallest. | '''(B)''' Intuitively, <math>x</math> will be largest for that option for which the value in the parentheses is smallest. | ||
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Thus the answer is <math>\boxed{\textbf{(B) } 3\left(1+\frac{r}{10}\right)^x = 7}</math>. | Thus the answer is <math>\boxed{\textbf{(B) } 3\left(1+\frac{r}{10}\right)^x = 7}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | As stated in the first solution, <math>x</math> will be largest when the value in the parenthesis is smallest. | ||
+ | |||
+ | We can plug in a value for <math>r</math> (for instance, <math>r=2</math>) and see which parenthesis has the smallest value. Doing so, we see that B is the answer. | ||
+ | |||
+ | -Made_in_2004 | ||
== See Also == | == See Also == |
Latest revision as of 16:58, 27 December 2020
Contents
Problem
Assume . Below are five equations for
. Which equation has the largest solution
?
Solution 1
(B) Intuitively, will be largest for that option for which the value in the parentheses is smallest.
Formally, first note that each of the values in parentheses is larger than . Now, each of the options is of the form
. This can be rewritten as
. As
, we have
. Thus
is the largest for the option for which
is smallest. And as
is an increasing function, this is the option for which
is smallest.
We now get the following easier problem: Given that , find the smallest value in the set
.
Clearly is smaller than the first and the third option.
We have , dividing both sides by
we get
.
And finally, , therefore
, and as both sides are positive, we can take the square root and get
.
Thus the answer is .
Solution 2
As stated in the first solution, will be largest when the value in the parenthesis is smallest.
We can plug in a value for (for instance,
) and see which parenthesis has the smallest value. Doing so, we see that B is the answer.
-Made_in_2004
See Also
2009 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 |
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