|
|
| (14 intermediate revisions by 6 users not shown) |
| Line 11: |
Line 11: |
| | </math> | | </math> |
| | | | |
| − | ==Solutions== | + | ==Solution 10 (Solution 1 but alternate)== |
| | | | |
| − | ===Solution 1=== | + | 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>. |
| − | In order to eliminate 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>
| |
| | | | |
| − | Given that <math>(\sqrt {49-x^2} - \sqrt {25-x^2}) = 3, (\sqrt {49-x^2} + \sqrt {25-x^2}) = \frac {24} {3} = \boxed{\textbf{(A) } 8}</math>. - cookiemonster2004
| + | ~Technodoggo |
| | | | |
| − | ===Solution 2=== | + | ==Video Solution (HOW TO THINK CREATIVELY!)== |
| − | 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{\textbf{(A) } 8}</math>.
| + | https://youtu.be/P-atxiiTw2I |
| | | | |
| − | ===Solution 3===
| + | ~Education, the Study of Everything |
| | | | |
| − | Put the equations to one side. <math>\sqrt{49-x^2}-\sqrt{25-x^2}=3</math> can be changed into <math>\sqrt{49-x^2}=\sqrt{25-x^2}+3</math>.
| |
| | | | |
| − | We can square both sides, getting us <math>49-x^2=(25-x^2)+(3^2)+ 2\cdot 3 \cdot \sqrt{25-x^2}.</math>
| |
| | | | |
| − | That simplifies out to <math>15=6 \sqrt{25-x^2}.</math> Dividing both sides by <math>6</math> gets us <math>\frac{5}{2}=\sqrt{25-x^2}</math>.
| |
| − |
| |
| − | Following that, we can square both sides again, resulting in the equation <math>\frac{25}{4}=25-x^2</math>. Simplifying that, we get <math>x^2 = \frac{75}{4}</math>.
| |
| − |
| |
| − | Substituting into the equation <math>\sqrt{49-x^2}+\sqrt{25-x^2}</math>, we get <math>\sqrt{49-\frac{75}{4}}+\sqrt{25-\frac{75}{4}}</math>. Immediately, we simplify into <math>\sqrt{\frac{121}{4}}+\sqrt{\frac{25}{4}}</math>. The two numbers inside the square roots are simplified to be <math>\frac{11}{2}</math> and <math>\frac{5}{2}</math>, so you add them up: <math>\frac{11}{2}+\frac{5}{2}=\boxed{\textbf{(A) 8}}</math>.
| |
| − |
| |
| − | ~kevinmathz
| |
| − |
| |
| − | ===Solution 4 (Geometric Interpretation)===
| |
| − |
| |
| − | Draw a right triangle <math>ABC</math> with a hypotenuse <math>AC</math> of length <math>7</math> and leg <math>AB</math> of length <math>x</math>. Draw <math>D</math> on <math>BC</math> such that <math>AD=5</math>. Note that <math>BC=\sqrt{49-x^2}</math> and <math>BD=\sqrt{25-x^2}</math>. Thus, from the given equation, <math>BC-BD=DC=3</math>. Using Law of Cosines on triangle <math>ADC</math>, we see that <math>\angle{ADC}=120^{\circ}</math> so <math>\angle{ADB}=60^{\circ}</math>. Since <math>ADB</math> is a <math>30-60-90</math> triangle, <math>\sqrt{25-x^2}=BD=\frac{5}{2}</math> and <math>\sqrt{49-x^2}=\frac{5}{2}+3=\frac{11}{2}</math>. Finally, <math>\sqrt{49-x^2}+\sqrt{25-x^2}=\frac{5}{2}+\frac{11}{2}=\boxed{\textbf{(A)~8}}</math>.
| |
| − | <asy>
| |
| − | var s = sqrt(3);
| |
| − | pair A = (-5*s/2, 0);
| |
| − | pair B = (0,0);
| |
| − | pair C = (0,5.5);
| |
| − | pair D = (0,2.5);
| |
| − |
| |
| − | draw(A--B--C--A--D);
| |
| − | rightanglemark(A, B, D);
| |
| − | label("A", A, SW);
| |
| − | label("B", B, SE);
| |
| − | label("C", C, NE);
| |
| − | label("D", D, E);
| |
| − | label("7", (-5*s/4, 5.5/2), NW);
| |
| − | label("120$^\circ$", D, NW);
| |
| − | label("60$^\circ$", (0,2), SW);
| |
| − | label("$x$", 0.5*A, S);
| |
| − | draw(rightanglemark(A, B, C));
| |
| − |
| |
| − | draw(anglemark(A, D, B));
| |
| − | markscalefactor = 0.04;
| |
| − | draw(anglemark(C, D, A));
| |
| − |
| |
| − | label("$\frac{5}{2}$", (0,1.25), E);
| |
| − | label("3", (0,4), E);
| |
| − | label("5", (-5*s/4, 5/4), N);
| |
| − | </asy>
| |
| − |
| |
| − | ===Solution 5 (No Square Roots, Fastest)===
| |
| − | We notice that the two expressions are conjugates, and therefore we can write them in a "difference-of-squares" format.
| |
| − | Namely, we can write it as <math>((49-x^2) - (25 - x^2)) = (49 - x^2 - 25 + x^2) = 24</math>. Given the <math>3</math> in the problem, we can divide <math>24 / 3 = \boxed{\textbf{(A) } 8}</math>.
| |
| − |
| |
| − | -aze.10
| |
| − |
| |
| − | ===Solution 6 (Symmetric Substitution)===
| |
| − | Since <math>\frac{25+49}{2}=37</math>, let <math>37-x^2 = y</math>. Then we have <math>\sqrt{y+12}-\sqrt{y-12}=3</math>. Squaring both sides gives us <math>2y-2\sqrt{y^2-144}=9</math>. Isolating the term with the square root, and squaring again, we get <math>4y^2-36y+81=4y^2-576 \implies y=\frac{73}{4}</math>. Then <math>\sqrt{y+12}+\sqrt{y-12} = \sqrt{\frac{121}{4}}+\sqrt{\frac{25}{4}} = \frac{16}{2}=\boxed{8}</math>.
| |
| − |
| |
| − | ===Solution 7 (Difference of Squares)===
| |
| − | Let <math>\sqrt{49-x^2}=a</math> and <math>\sqrt{25-x^2}=b</math>. Then by difference of squares:
| |
| − |
| |
| − | <math>(a+b)(a-b)=a^2-b^2</math>.
| |
| − |
| |
| − | We can simplify this expression to get our answer. <math>a^2-b^2=(49-x^2)-(25-x^2)=24</math> and from the given statement, <math>a-b=3</math>. Now we have:
| |
| − |
| |
| − | <math>(a+b)(3)=24</math>.
| |
| − |
| |
| − | Hence, <math>a+b=\sqrt{49-x^2}-\sqrt{25-x^2}=8</math> so our answer is <math>\boxed{\textbf{(A) } 8}</math>.
| |
| − |
| |
| − | ~BakedPotato66
| |
| − |
| |
| − | ===Solution 8 (Analytic Geometry)===
| |
| − |
| |
| − | [[File:2018 AMC10 A P10.PNG|500px]]
| |
| − |
| |
| − | ~isabelchen
| |
| | | | |
| | == Video Solutions == | | == Video Solutions == |
satisfies ![\[\sqrt{49-x^2}-\sqrt{25-x^2}=3\]](http://latex.artofproblemsolving.com/6/0/2/60260471de484d4c7c3af264a757d4ff5de2ad3d.png)
What is the value of 
?
; in other words, we want to find 
. We know that 
Thus, 
.
