Answer to Question #198834 in Real Analysis for owais afzal

Question

A small rectangular warehouse is to be constructed which is to have an area of 10000 square feet. The building is to be partitioning internally in to eight equal parts. The costs have been estimated based on exterior and interior walls dimensions. The costs are $200 per running foot of exterior wall and plus $100 per running foot of interior wall.

Answer 1

Length of warehouse = $L$

Breadth of warehouse = $B$

Area of warehouse = $L\times B=10000$

$L=\dfrac{10^{4}}{B}$

Cost of painting outer walls = $C_1=200\times2[2(L+B)]$

Cost of painting interior walls = $C_2=100\times2[L+3B]$

Total Cost of painting = $C=C_1+C_2$

$C=800L+800B+200L+600B$

$C=1000L+1400B$

$C=10^7B^{-1}+1400B$

Differentiating above equation,

$C'=-10^{7}B^{-2}+1400=0$

$B^2=\dfrac{10^7}{1400}$

$B=84.5\space ft$

$C''(84.5)=2\times10^7B^{-3}=2\times10^7(84.5)^{-3}>0(minima)$

$L=118.34\space ft$

Therefore, for painting costs to be minimum the dimensions of rectangular warehouse should be

$L=118.34\space ft$

$B=84.5\space ft$