Welcome to our community

Be a part of something great, join today!

Problem Of The Week #464 April 19th 2021

Status
Not open for further replies.
  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
  • Thread starter
  • Admin
  • #2

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
No one answered last week's POTW. (Sadface) However, you can find the suggested model solution as follows:
From the conditions of the problem we obtain

$\begin{align*} 252=3x+7y+14z& \ge 3\sqrt[3]{3x(7y)(14z)}\\&=3\sqrt[3]{3(7)(14)(2016+u^2)}\\& \ge 3\sqrt[3]{3(7)(14)(2016)}\\&=3\sqrt[3]{2^6(3^3)(7^3)}\\&=2^2(3^2)(7)\\&=252\end{align*}$

Equality is attainable when $3x=7y=14z$ and $u=0$. From the first equation of the system, we obtain

$3x=7y=14z=\dfrac{252}{3}=84$. This implies $x=28,\,y=12$ and $z=6$.
 
Status
Not open for further replies.