If j, k, and l are positive with jk = 24, jl = 48, and kl = 18, find j + k + l.
If j, k, and l are positive with jk = 24, jl = 48, and kl = 18, find j + k + l.
jk = 24 ⇒ 24/j = k
jl = 48 ⇒ 48/j = l
kl = 18
So
kl = 18
(24/j) (48/j) = 18
1152 / j^2 = 18
j^2 = 1152/18
j^2 = 64 take the positive root
j = 8
So.....24 / j = k ⇒ 24 / 8 = k ...... 3 = k
And 48 / j = l ⇒ 48 / 8 = l ...... 6 = l
So
j + k + l = 8 + 3 + 6 = 17
If j, k, and l are positive with jk = 24, jl = 48, and kl = 18,
find j + k + l.
1.
\(\begin{array}{|rcll|} \hline jk\cdot jl \cdot kl = (jkl)^2 &=& 24\cdot 48\cdot 18 \\ &=& 20736 \\ jkl &=& \sqrt{20736} \\ \mathbf{jkl} &\mathbf{=}& \mathbf{ 144} \\ \hline \end{array}\)
2.
\(\begin{array}{|rcll|} \hline j+k+l &=& \dfrac{jk\cdot jl}{jkl} + \dfrac{jk\cdot kl}{jkl} + \dfrac{jl\cdot kl}{jkl} \\ &=& \dfrac{jk\cdot jl+jk\cdot kl+jl\cdot kl}{jkl} \\ &=& \dfrac{24\cdot 48+24\cdot 18+48\cdot 18}{144} \\ \mathbf{j+k+l} &\mathbf{=}& \mathbf{ 17} \\ \hline \end{array}\)