2015 AIME II Problems/Problem 4: Difference between revisions

Revision as of 13:04, 14 February 2023

Problem

In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$ , and the altitude to these bases has length $\log 16$ . The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$ , where $p$ and $q$ are positive integers. Find $p + q$ .

Solution

Call the trapezoid $ABCD$ with $AB$ as the smaller base and $CD$ as the longer. Let the point where an altitude intersects the larger base be $E$ , where $E$ is closer to $D$ .

Subtract the two bases and divide to find that $ED$ is $\log 8$ . The altitude can be expressed as $\frac{4}{3} \log 8$ . Therefore, the two legs are $\frac{5}{3} \log 8$ , or $\log 32$ .

The perimeter is thus $\log 32 + \log 32 + \log 192 + \log 3$ which is $\log 2^{16} 3^2$ . So $p + q = \boxed{018}$

Solution 2 (gratuitous wishful thinking)

Set the base of the log as 2. Then call the trapezoid $ABCD$ with $CD$ as the longer base. Then have the two feet of the altitudes be $E$ and $F$ , with $E$ and $F$ in position from left to right respectively. Then, $CF$ and $ED$ are $\log 192 - \log 3 = \log 64$ (from the log subtraction identity. Then $CF=EF=3$ (isosceles trapezoid and $\log 64$ being 6. Then the 2 legs of the trapezoid is $\sqrt{3^2+4^2}=5=\log 32$ .

And we have the answer:

$\log 192 + \log 32 + \log 32 + \log 3 = \log(192 \cdot 32 \cdot 32 \cdot 3) = \log(2^6 \cdot 3 \cdot 2^5 \cdot 2^5 \cdot 3) = \log(2^{16} \cdot 3^2) \Rightarrow 16+2 = \boxed{18}$

-dragoon

Video Solution

https://www.youtube.com/watch?v=9re2qLzOKWk&t=226s

~MathProblemSolvingSkills.com

@@ Line 21: / Line 21: @@
 -dragoon
+==Video Solution==
+https://www.youtube.com/watch?v=9re2qLzOKWk&t=226s
+~MathProblemSolvingSkills.com
 ==See also==
 {{AIME box|year=2015|n=II|num-b=3|num-a=5}}
 {{MAA Notice}}

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