Difference between revisions of "2018 AMC 10A Problems/Problem 10"
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~ SAMANTAP | ~ SAMANTAP | ||
+ | ==Solution 4== | ||
+ | Move <math>-\sqrt{25-x^2}</math> to the right to get <math>\sqrt{49-x^2} = 3 + \sqrt{25-x^2}</math>. | ||
+ | Square both sides to get <math>49-x^2 = 9 + 6\sqrt{25-x^2} + (25-x^2)</math>. | ||
+ | Simplify to get <math>15 = 6\sqrt{25-x^2}</math>, or <math>\frac{5}{2} = \sqrt{25-x^2}</math> | ||
+ | Substitute this back into the original equation tog et that <math>\sqrt{49-x^2} = \frac{11}{2}</math>. The answer is <math>\boxed{\textbf{(A) }8}</math> | ||
+ | |||
+ | -Failure.net | ||
==Video Solution (HOW TO THINK CREATIVELY!)== | ==Video Solution (HOW TO THINK CREATIVELY!)== | ||
https://youtu.be/P-atxiiTw2I | https://youtu.be/P-atxiiTw2I |
Revision as of 14:13, 23 July 2024
Contents
[hide]Problem
Suppose that real number satisfies What is the value of ?
Solution 1
We let ; in other words, we want to find . We know that Thus, .
~Technodoggo
Solution 2
Let , and . Solving for the constants in terms of x, a , and b, we get , and . Subtracting the second equation from the first gives us . Difference of squares gives us . Since we want to find , and we know , we get , so
~idk12345678
Solution 3
We can substitute for , thus turning the equation into . Moving the to the other side and squaring gives us , solving for gives us 25/4. We substitute this value into the expression they asked us to evaluate giving 8.
~ SAMANTAP
Solution 4
Move to the right to get . Square both sides to get . Simplify to get , or Substitute this back into the original equation tog et that . The answer is
-Failure.net
Video Solution (HOW TO THINK CREATIVELY!)
~Education, the Study of Everything
Video Solutions
Video Solution 1
https://youtu.be/ba6w1OhXqOQ?t=1403
~ pi_is_3.14
Video Solution 2
https://youtu.be/zQG70XKAdeA ~ North America Math Contest Go Go Go
Video Solution 3
Video Solution 4
~savannahsolver
See Also
2018 AMC 10A (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 |