Law of Cosines and to Find Extremum by Completing Square

 In $\displaystyle \Delta ABC$, $\displaystyle AB=(5-x)$ cm, $\displaystyle BC=(4+x)$ cm, $\displaystyle \angle AsBC=120{}^\circ$ and $\displaystyle AC=y$ cm.

(a)     Show that $\displaystyle {{y}^{2}}={{x}^{2}}-x+61$.

(b)     Find the minimum value of $\displaystyle {{y}^{2}}$, and give the value of $\displaystyle x$ for which this occurs.

Solution

          $\displaystyle AB=(5-x)$ cm,  
          $\displaystyle BC=(4+x)$ cm, 
          $\displaystyle \angle ABC=120{}^\circ$ 
          $\displaystyle AC=y$ cm. 

(a)      By the law of cosines, 

          $\displaystyle A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}-2\cdot AB\cdot AC\cos (\angle ABC)$ 
      
          $\displaystyle {{y}^{2}}={{(5-x)}^{2}}+{{(4+x)}^{2}}-2(5-x)(4+x)\cos 120{}^\circ$

          $\displaystyle {{y}^{2}}=25-10x+{{x}^{2}}+16+8x+{{x}^{2}}+2(5-x)(4+x)\left( {\frac{1}{2}} \right)$

          $\displaystyle {{y}^{2}}=41-2x+2{{x}^{2}}-{{x}^{2}}+x+20$

          $\displaystyle {{y}^{2}}={{x}^{2}}-x+61$

      $\displaystyle \therefore {{y}^{2}}={{x}^{2}}-x+\frac{1}{4}+61-\frac{1}{4}$

      $\displaystyle \therefore {{y}^{2}}={{\left( {x-\frac{1}{2}} \right)}^{2}}+60.75$

          Since $\displaystyle {{\left( {x-\frac{1}{2}} \right)}^{2}}\ge 0\ $ for all $\displaystyle x\in R$,

          $\displaystyle {{\left( {x-\frac{1}{2}} \right)}^{2}}+60.75\ge 60.75$

      $\displaystyle \therefore {{y}^{2}}\ge 60.75$.

          Therefore the minimum value of $\displaystyle {{y}^{2}}$ is $\displaystyle 60.75$ and this value occurs when $\displaystyle x=\frac{1}{2}$.     

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