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Difference between revisions of "1963 AHSME Problems/Problem 40"

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== Problem 40==
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== Problem ==
  
 
If <math>x</math> is a number satisfying the equation <math>\sqrt[3]{x+9}-\sqrt[3]{x-9}=3</math>, then <math>x^2</math> is between:
 
If <math>x</math> is a number satisfying the equation <math>\sqrt[3]{x+9}-\sqrt[3]{x-9}=3</math>, then <math>x^2</math> is between:
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\textbf{(E)}\ 95\text{ and }105 </math>
 
\textbf{(E)}\ 95\text{ and }105 </math>
  
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==Solution==
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==Solution 1==
  
 
Let <math>a = \sqrt[3]{x + 9}</math> and <math>b = \sqrt[3]{x - 9}</math>. Cubing these equations, we get <math>a^3 = x + 9</math> and <math>b^3 = x - 9</math>, so <math>a^3 - b^3 = 18</math>. The left-hand side factors as
 
Let <math>a = \sqrt[3]{x + 9}</math> and <math>b = \sqrt[3]{x - 9}</math>. Cubing these equations, we get <math>a^3 = x + 9</math> and <math>b^3 = x - 9</math>, so <math>a^3 - b^3 = 18</math>. The left-hand side factors as
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Squaring the equation <math>a - b = 3</math>, we get <math>a^2 - 2ab + b^2 = 9</math>. Subtracting this equation from the equation <math>a^2 + ab + b^2 = 6</math>, we get <math>3ab = -3</math>, so <math>ab = -1</math>. But <math>a = \sqrt[3]{x + 9}</math> and <math>b = \sqrt[3]{x - 9}</math>, so <math>ab = \sqrt[3]{(x + 9)(x - 9)} = \sqrt[3]{x^2 - 81}</math>, so <math>\sqrt[3]{x^2 - 81} = -1</math>. Cubing both sides, we get <math>x^2 - 81 = -1</math>, so <math>x^2 = 80</math>. The answer is <math>\boxed{\textbf{(C)}}</math>.
 
Squaring the equation <math>a - b = 3</math>, we get <math>a^2 - 2ab + b^2 = 9</math>. Subtracting this equation from the equation <math>a^2 + ab + b^2 = 6</math>, we get <math>3ab = -3</math>, so <math>ab = -1</math>. But <math>a = \sqrt[3]{x + 9}</math> and <math>b = \sqrt[3]{x - 9}</math>, so <math>ab = \sqrt[3]{(x + 9)(x - 9)} = \sqrt[3]{x^2 - 81}</math>, so <math>\sqrt[3]{x^2 - 81} = -1</math>. Cubing both sides, we get <math>x^2 - 81 = -1</math>, so <math>x^2 = 80</math>. The answer is <math>\boxed{\textbf{(C)}}</math>.
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==Solution 2==
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<math>\sqrt[3]{x+9}-\sqrt[3]{x-9}-3=0</math> i.e, <math>\sqrt[3]{x+9}+\sqrt[3]{-x+9}+(-3)=0</math>
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if the sum of three numbers is zero, then their sum of cubes is thrice the product of each number.
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then, <math>x+9-x+9+27=3(\sqrt[3]{x+9})(\sqrt[3]{9-x})(-3)</math> . by solving this, we get <math>-9=-9\sqrt[3]{9^2-x^2}</math>
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<math>x^2=80</math>. this gives the step what we had done in solution 1.The answer is <math>\boxed{\textbf{(C)}}</math>.
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==Solution 3==
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Cubing both sides, we get
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<cmath>x+9-3\sqrt[3]{(x+9)^2(x-9)}+3\sqrt[3]{(x+9)(x-9)^2}-x+9=27,</cmath>so <cmath>18-3\sqrt[3]{(x-9)(x+9)}(\sqrt[3]{x+9}-\sqrt[3]{x-9})=27\implies 3\sqrt[3]{x^2-81}=-3.</cmath> Dividing both sides by 3 and cubing, we find <math>x^2=80</math>, which is between <math>\boxed{(C)75\text{ and }85}</math>.
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~pfalcon
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==Solution 4==
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We consider the formula <math>(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3</math>. Factoring out <math>3ab</math>, and the commutative property gives <math>( 
 +
a^3 - b^3) - 3ab(a-b)</math>. Cubing both sides gives us
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<cmath>(\sqrt[3]{x+9}-\sqrt[3]{x-9})^3=27</cmath>
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Using our formula, we have
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<cmath>18 - 9 \sqrt[3]{x^2 - 81} = 27</cmath>
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Solving this gives <math>x^2 = 80</math>, therefore, the answer is <math>\boxed{\textbf{(C)}}</math>.
 +
~zixuan12
  
 
==See Also==
 
==See Also==

Latest revision as of 13:03, 26 March 2023

Problem

If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between:

$\textbf{(A)}\ 55\text{ and }65\qquad \textbf{(B)}\ 65\text{ and }75\qquad \textbf{(C)}\ 75\text{ and }85\qquad \textbf{(D)}\ 85\text{ and }95\qquad \textbf{(E)}\ 95\text{ and }105$

Solution 1

Let $a = \sqrt[3]{x + 9}$ and $b = \sqrt[3]{x - 9}$. Cubing these equations, we get $a^3 = x + 9$ and $b^3 = x - 9$, so $a^3 - b^3 = 18$. The left-hand side factors as \[(a - b)(a^2 + ab + b^2) = 18.\]

However, from the given equation $\sqrt[3]{x + 9} - \sqrt[3]{x - 9} = 3$, we get $a - b = 3$. Then $3(a^2 + ab + b^2) = 18$, so $a^2 + ab + b^2 = 18/3 = 6$.

Squaring the equation $a - b = 3$, we get $a^2 - 2ab + b^2 = 9$. Subtracting this equation from the equation $a^2 + ab + b^2 = 6$, we get $3ab = -3$, so $ab = -1$. But $a = \sqrt[3]{x + 9}$ and $b = \sqrt[3]{x - 9}$, so $ab = \sqrt[3]{(x + 9)(x - 9)} = \sqrt[3]{x^2 - 81}$, so $\sqrt[3]{x^2 - 81} = -1$. Cubing both sides, we get $x^2 - 81 = -1$, so $x^2 = 80$. The answer is $\boxed{\textbf{(C)}}$.


Solution 2

$\sqrt[3]{x+9}-\sqrt[3]{x-9}-3=0$ i.e, $\sqrt[3]{x+9}+\sqrt[3]{-x+9}+(-3)=0$ 

if the sum of three numbers is zero, then their sum of cubes is thrice the product of each number.
then, $x+9-x+9+27=3(\sqrt[3]{x+9})(\sqrt[3]{9-x})(-3)$ . by solving this, we get $-9=-9\sqrt[3]{9^2-x^2}$
$x^2=80$. this gives the step what we had done in solution 1.The answer is $\boxed{\textbf{(C)}}$.

Solution 3

Cubing both sides, we get \[x+9-3\sqrt[3]{(x+9)^2(x-9)}+3\sqrt[3]{(x+9)(x-9)^2}-x+9=27,\]so \[18-3\sqrt[3]{(x-9)(x+9)}(\sqrt[3]{x+9}-\sqrt[3]{x-9})=27\implies 3\sqrt[3]{x^2-81}=-3.\] Dividing both sides by 3 and cubing, we find $x^2=80$, which is between $\boxed{(C)75\text{ and }85}$.

~pfalcon

Solution 4

We consider the formula $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$. Factoring out $3ab$, and the commutative property gives $( a^3 - b^3) - 3ab(a-b)$. Cubing both sides gives us \[(\sqrt[3]{x+9}-\sqrt[3]{x-9})^3=27\] Using our formula, we have \[18 - 9 \sqrt[3]{x^2 - 81} = 27\] Solving this gives $x^2 = 80$, therefore, the answer is $\boxed{\textbf{(C)}}$. ~zixuan12

See Also

1963 AHSC (Problems • Answer Key • Resources)
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