Difference between revisions of "1961 AHSME Problems/Problem 40"
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| − | ==Problem 40== | + | == Problem 40== |
Find the minimum value of <math>\sqrt{x^2+y^2}</math> if <math>5x+12y=60</math>. | Find the minimum value of <math>\sqrt{x^2+y^2}</math> if <math>5x+12y=60</math>. | ||
| − | <math>\textbf{(A) }\frac{60}{13}\qquad\textbf{(B) }\frac{13}{5} \qquad\textbf{(C) }\frac{13}{12}\qquad\textbf{(D) }1 \qquad\textbf{(E) }0</math> | + | <math>\textbf{(A)}\ \frac{60}{13}\qquad |
| + | \textbf{(B)}\ \frac{13}{5}\qquad | ||
| + | \textbf{(C)}\ \frac{13}{12}\qquad | ||
| + | \textbf{(D)}\ 1\qquad | ||
| + | \textbf{(E)}\ 0 </math> | ||
| + | |||
| + | ==Solutions (WIP)== | ||
| + | |||
| + | ===Solution 1=== | ||
| + | Solve for <math>y</math> in the linear equation. | ||
| + | <cmath>12y = 60 - 5x</cmath> | ||
| + | <cmath>y = 5 - \frac{5x}{12}</cmath> | ||
| + | Substitute <math>y</math> in <math>\sqrt{x^2+y^2}</math>. | ||
| + | <cmath>\sqrt{x^2 + (5 - \frac{5x}{12})^2}</cmath> | ||
| + | <cmath>\sqrt{x^2 + 25 - \frac{25x}{6} + \frac{25x^2}{144}}</cmath> | ||
| + | <cmath>\sqrt{\frac{169x^2}{144} - \frac{25x}{6} + 25}</cmath> | ||
| + | |||
| + | ===Solution 2=== | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME 40p box|year=1961|num-b=26|num-a=28}} | ||
| + | |||
| + | [[Category:Intermediate Algebra Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 10:56, 25 May 2018
Problem 40
Find the minimum value of 
if 
.
Solutions (WIP)
Solution 1
Solve for 
in the linear equation.
![\[12y = 60 - 5x\]](http://latex.artofproblemsolving.com/4/5/0/450de5e724868bf9cc9c974f8bb752931c80a71c.png)
![\[y = 5 - \frac{5x}{12}\]](http://latex.artofproblemsolving.com/d/4/7/d47913abdeca3e89723c2d9985dc93bc7a2506fe.png)
Substitute in .
![\[\sqrt{x^2 + (5 - \frac{5x}{12})^2}\]](http://latex.artofproblemsolving.com/3/f/2/3f28484b0848e8acbffbd429c2c7e088ce1cd43f.png)
![\[\sqrt{x^2 + 25 - \frac{25x}{6} + \frac{25x^2}{144}}\]](http://latex.artofproblemsolving.com/c/1/5/c151c164a18d008ca584863fa12f42b5b7e8f962.png)
![\[\sqrt{\frac{169x^2}{144} - \frac{25x}{6} + 25}\]](http://latex.artofproblemsolving.com/6/c/8/6c89cb52d6905e0396434eca4b5fa8abb1acad14.png)
Solution 2
See Also
| 1961 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 26 |
Followed by Problem 28 | |
| 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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
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