Menu Close

Find-maximum-value-of-the-product-xy-72-3x-4y-for-positive-value-of-x-amp-y-




Question Number 141164 by iloveisrael last updated on 16/May/21
    Find maximum value      of the product xy(72−3x−4y)      for positive value of x & y.
$$\:\:\:\:{Find}\:{maximum}\:{value}\: \\ $$$$\:\:\:{of}\:{the}\:{product}\:{xy}\left(\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\right) \\ $$$$\:\:\:\:{for}\:{positive}\:{value}\:{of}\:{x}\:\&\:{y}. \\ $$
Commented by mitica last updated on 16/May/21
max=1152 for x=8,y=6
$${max}=\mathrm{1152}\:{for}\:{x}=\mathrm{8},{y}=\mathrm{6} \\ $$
Answered by EDWIN88 last updated on 16/May/21
The terms of the product x,y & 72−3x−4y  do not have a constant sum. However if we  adjust the product and write it as           (1/(12))(3x)(4y)(72−3x−4y)    then ignoring the multiplier (1/(12)) for a moment  we see that the factors 3x,4y & 72−3x−4y   have a constant 72. Thus we seek value of   x & y satisfying          3x=4y=72−3x−4y   given the unique values  { ((x=8)),((y=6)) :}   Thus the max value of xy(72−3x−4y) is         8×6×24 = 1,152 ⋇
$$\mathrm{The}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{product}\:\mathrm{x},\mathrm{y}\:\&\:\mathrm{72}−\mathrm{3x}−\mathrm{4y} \\ $$$$\mathrm{do}\:\mathrm{not}\:\mathrm{have}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{sum}.\:\mathrm{However}\:\mathrm{if}\:\mathrm{we} \\ $$$$\mathrm{adjust}\:\mathrm{the}\:\mathrm{product}\:\mathrm{and}\:\mathrm{write}\:\mathrm{it}\:\mathrm{as}\: \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{12}}\left(\mathrm{3x}\right)\left(\mathrm{4y}\right)\left(\mathrm{72}−\mathrm{3x}−\mathrm{4y}\right)\: \\ $$$$\:\mathrm{then}\:\mathrm{ignoring}\:\mathrm{the}\:\mathrm{multiplier}\:\frac{\mathrm{1}}{\mathrm{12}}\:\mathrm{for}\:\mathrm{a}\:\mathrm{moment} \\ $$$$\mathrm{we}\:\mathrm{see}\:\mathrm{that}\:\mathrm{the}\:\mathrm{factors}\:\mathrm{3x},\mathrm{4y}\:\&\:\mathrm{72}−\mathrm{3x}−\mathrm{4y} \\ $$$$\:\mathrm{have}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{72}.\:\mathrm{Thus}\:\mathrm{we}\:\mathrm{seek}\:\mathrm{value}\:\mathrm{of} \\ $$$$\:\mathrm{x}\:\&\:\mathrm{y}\:\mathrm{satisfying}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{3x}=\mathrm{4y}=\mathrm{72}−\mathrm{3x}−\mathrm{4y} \\ $$$$\:\mathrm{given}\:\mathrm{the}\:\mathrm{unique}\:\mathrm{values}\:\begin{cases}{\mathrm{x}=\mathrm{8}}\\{\mathrm{y}=\mathrm{6}}\end{cases} \\ $$$$\:\mathrm{Thus}\:\mathrm{the}\:\mathrm{max}\:\mathrm{value}\:\mathrm{of}\:\mathrm{xy}\left(\mathrm{72}−\mathrm{3x}−\mathrm{4y}\right)\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\mathrm{8}×\mathrm{6}×\mathrm{24}\:=\:\mathrm{1},\mathrm{152}\:\divideontimes \\ $$
Commented by iloveisrael last updated on 16/May/21
Yes....
$$\mathrm{Y}{es}…. \\ $$
Answered by mitica last updated on 16/May/21
if 72−3x−4y≤0⇒xy(72−3x−4y)≤0  if 72−3x−4y>0⇒(1/(12))∙3x∙4y∙(72−3x−4y)≤^(am−gm) (1/(12))(3x+4y−72−3x−4y)^3 ∙(1/(27))=1152  = for 3x=4y=72−3x−4y⇒x=8,y=6  ⇒max{xy(72−3x−4y)}=1152
$${if}\:\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\leqslant\mathrm{0}\Rightarrow{xy}\left(\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\right)\leqslant\mathrm{0} \\ $$$${if}\:\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}>\mathrm{0}\Rightarrow\frac{\mathrm{1}}{\mathrm{12}}\centerdot\mathrm{3}{x}\centerdot\mathrm{4}{y}\centerdot\left(\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\right)\overset{{am}−{gm}} {\leqslant}\frac{\mathrm{1}}{\mathrm{12}}\left(\mathrm{3}{x}+\mathrm{4}{y}−\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\right)^{\mathrm{3}} \centerdot\frac{\mathrm{1}}{\mathrm{27}}=\mathrm{1152} \\ $$$$=\:{for}\:\mathrm{3}{x}=\mathrm{4}{y}=\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\Rightarrow{x}=\mathrm{8},{y}=\mathrm{6} \\ $$$$\Rightarrow{max}\left\{{xy}\left(\mathrm{72}−\mathrm{3}{x}−\mathrm{4}{y}\right)\right\}=\mathrm{1152} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *