Question

Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60.$

$\dfrac{60}{13}$

$\dfrac{13}{5}$

$\dfrac{13}{12}$

$1$

$0$

Answer: A

Concepts:distance formula Cauchy-Schwarz Inequality optimization

Difficulty rating: 1300

Small Hint:

Interpret $\sqrt{x^2+y^2}$ as distance from the origin

Big Hint:

Apply Cauchy-Schwarz to $5x+12y$

Solution:

By Cauchy-Schwarz, $\begin{aligned} 60&=5x+12y\\ &\le\sqrt{5^2+12^2}\\ &\qquad\cdot\sqrt{x^2+y^2}\\ &=13\sqrt{x^2+y^2}. \end{aligned}$ Thus the distance is at least $\frac{60}{13}.$ Equality occurs when $(x,y)$ is proportional to $(5,12),$ so the minimum is $\frac{60}{13}.$

Therefore, the correct answer is A.

1961 AMC 12 Problem 40

Answer: A

Difficulty rating: 1300

Small Hint:

Big Hint:

Solution:

Problem 40 in Other Years