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

Revision as of 11:25, 19 June 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 1

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

Solution 2

Let $a = \sqrt{49-x^2}$ , and $b = \sqrt{25-x^2}$ . Solving for the constants in terms of x, a , and b, we get $a^2 + x^2 = 49$ , and $b^2 + x^2 = 25$ . Subtracting the second equation from the first gives us $a^2 - b^2 = 24$ . Difference of squares gives us $(a+b)(a-b) = 24$ . Since we want to find $a+b = \sqrt{49-x^2}+\sqrt{25-x^2}$ , and we know $a-b = 3$ , we get $3(a+b) = 24$ , so $a+b = \boxed{\textbf{(A) }8}$

~idk12345678

Solution 3

Let $a = 25 - x^2$ , so $\sqrt{a+24}-\sqrt{a}=3$ , we add the $\sqrt{a} to the other side$ to make $\sqrt{a+24}=3+\sqrt{a}$ . Sqauring both sides gives us, $a + 24 = 9 + 6\sqrt{a} + a$ , so $6\sqrt{a}=15$ , and $a = 25/4$ , substituting this value for the equation they are asking us to solve gives us 8.

- samantap

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

@@ Line 24: / Line 24: @@
 ==Solution 3==
+Let <math>a = 25 - x^2</math>, so <math>\sqrt{a+24}-\sqrt{a}=3</math>, we add the <math>\sqrt{a} to the other side</math> to make <math>\sqrt{a+24}=3+\sqrt{a}</math>. Sqauring both sides gives us, <math>a + 24 = 9 + 6\sqrt{a} + a</math>, so <math>6\sqrt{a}=15</math>, and <math>a = 25/4</math>, substituting this value for the equation they are asking us to solve gives us 8.
+- samantap
 ==Video Solution (HOW TO THINK CREATIVELY!)==

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

Page

Toolbox

Search