Difference between revisions of "2018 AMC 10A Problems/Problem 10"

Latest revision as of 19:59, 20 April 2024

Problem

Suppose that real number $x$ satisfies $\sqrt{49-x^2}-\sqrt{25-x^2}=3$ What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$ ?

$\textbf{(A) }8\qquad \textbf{(B) }\sqrt{33}+8\qquad \textbf{(C) }9\qquad \textbf{(D) }2\sqrt{10}+4\qquad \textbf{(E) }12\qquad$

Solution 10 (Solution 1 but alternate)

We let $a=\sqrt{49-x^2}+\sqrt{25-x^2}$ ; in other words, we want to find $a$ . We know that $a\cdot3=\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\cdot\left(\sqrt{49-x^2}-\sqrt{25-x^2}\right)=\left(\sqrt{49-x^2}\right)^2-\left(\sqrt{25-x^2}\right)^2=\left(49-x^2\right)-\left(25-x^2\right)=24.$ Thus, $a=\boxed{8}$ .

~Technodoggo

Video Solution (HOW TO THINK CREATIVELY!)

https://youtu.be/P-atxiiTw2I

~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

https://youtu.be/ZiZVIMmo260

Video Solution 4

https://youtu.be/5cA87rbzFdw

~savannahsolver

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

@@ Line 1: / Line 1: @@
-Suppose that real number <math>x</math> satisfies <cmath>\sqrt{49-x^2}-\sqrt{25-x^2}=3</cmath>. What is the value of <math>\sqrt{49-x^2}+\sqrt{25-x^2}</math>?
+==Problem==
+Suppose that real number <math>x</math> satisfies <cmath>\sqrt{49-x^2}-\sqrt{25-x^2}=3</cmath>What is the value of <math>\sqrt{49-x^2}+\sqrt{25-x^2}</math>?
 <math>
-\textbf{(A) }8 \qquad
+\textbf{(A) }8\qquad
 \textbf{(B) }\sqrt{33}+8\qquad
-\textbf{(C) }9 \qquad
+\textbf{(C) }9\qquad
-\textbf{(D) }2\sqrt{10}+4 \qquad
+\textbf{(D) }2\sqrt{10}+4\qquad
-\textbf{(E) }12 \qquad
+\textbf{(E) }12\qquad
 </math>
-==Solutions==
-=== Solution 1===
-In order to get rid of the square roots, we multiply by the conjugate. Its value is the solution.The <math>x^2</math> terms cancel nicely. <math>(\sqrt {49-x^2} + \sqrt {25-x^2}) * (\sqrt {49-x^2} - \sqrt {25-x^2}) = 49-x^2 - 25 +x^2 = 24</math>
+==Solution 10 (Solution 1 but alternate)==
+We let <math>a=\sqrt{49-x^2}+\sqrt{25-x^2}</math>; in other words, we want to find <math>a</math>. We know that <math>a\cdot3=\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\cdot\left(\sqrt{49-x^2}-\sqrt{25-x^2}\right)=\left(\sqrt{49-x^2}\right)^2-\left(\sqrt{25-x^2}\right)^2=\left(49-x^2\right)-\left(25-x^2\right)=24.</math> Thus, <math>a=\boxed{8}</math>.
+~Technodoggo
+==Video Solution (HOW TO THINK CREATIVELY!)==
+https://youtu.be/P-atxiiTw2I
+~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
-Given that <math>(\sqrt {49-x^2} - \sqrt {25-x^2})</math> = 3, <math>(\sqrt {49-x^2} + \sqrt {25-x^2}) = \frac {24} {3} = \boxed{(A) 8}</math>
+===Video Solution 3===
+https://youtu.be/ZiZVIMmo260
-Solution by PancakeMonster2004, explanations added by a1b2.
+===Video Solution 4===
-===Solution 2 (bad)===
+https://youtu.be/5cA87rbzFdw
-Let <math>u=\sqrt{49-x^2}</math>, and let <math>v=\sqrt{25-x^2}</math>. Then <math>v=\sqrt{u^2-24}</math>. Substituting, we get <math>u-\sqrt{u^2-24}=3</math>. Rearranging, we get <math>u-3=\sqrt{u^2-24}</math>. Squaring both sides and solving, we get <math>u=\frac{11}{2}</math> and <math>v=\frac{11}{2}-3=\frac{5}{2}</math>. Adding, we get that the answer is <math>\boxed{(A) 8}</math>
-== See Also ==
+~savannahsolver
+==See Also==
 {{AMC10 box|year=2018|ab=A|num-b=9|num-a=11}}
-{{MAA Notice}}
+[[Category:Introductory Algebra Problems]]

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