Difference between revisions of "2001 AMC 10 Problems/Problem 10"
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<math>\textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24</math> | <math>\textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24</math> | ||
− | == Solution == | + | == Solution 1== |
Look at the first two equations in the problem. | Look at the first two equations in the problem. | ||
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Since we know every variable's value, we can substitute it in for <math> x+y+z = 4+6+12 = \boxed{\textbf{(D) }22} </math>. | Since we know every variable's value, we can substitute it in for <math> x+y+z = 4+6+12 = \boxed{\textbf{(D) }22} </math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | These equations are symmetric, and furthermore, they use multiplication. This makes us think to multiply them all. This gives <math>(xy)(yz)(xz) = (xyz)^2 = (24)(48)(72) = (24 \times 12)^2</math>. | ||
+ | We square root: | ||
+ | <math>xyz = 288</math>. | ||
+ | Aha! We divide each of the given equations into this, yielding <math>x = 4</math>, <math>y = 6</math>, and <math>z = 12</math>. The desired sum is <math>4+6+12 = \boxed{22}</math>, so the answer is <math>\boxed{\text{(D)}}</math>. | ||
== See Also == | == See Also == | ||
{{AMC10 box|year=2001|num-b=9|num-a=11}} | {{AMC10 box|year=2001|num-b=9|num-a=11}} |
Revision as of 21:32, 1 June 2011
Contents
[hide]Problem
If , , and are positive with , , and , then is
Solution 1
Look at the first two equations in the problem.
and .
We can say that .
Given , we can substitute for and find
.
We can replace y into the first equation. .
Since we know every variable's value, we can substitute it in for .
Solution 2
These equations are symmetric, and furthermore, they use multiplication. This makes us think to multiply them all. This gives . We square root: . Aha! We divide each of the given equations into this, yielding , , and . The desired sum is , so the answer is .
See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 10 Problems and Solutions |