Question 1099839
The container does not have a top.
 The base and sides are rectangular. 
The width is x cm. The length is 4 times the width.
 The volume is 600 cm^3.
 Determine the minimum surface area to two decimal places. 
:
let h = the height of the box
Given
x = the width
4x = the length
Volume
4x * x * h = 600
4x^2 * h = 600
h = {{{600/(4x^2)}}}
cancel the 4
h = {{{120/x^2}}}
:
Surface area
S.A. =(4x*x) + 2(4x*h) + 2(x*h)
S.A. = 4x^2 + 8xh + 2xh
S.A. = 4x^2 + 10xh
replace h with {{{120/x^2}}}
S.A. = 4x^2 + 10x*{{{120/x^2}}}
Cancel x
S.A. = 4x^2 + {{{1200/x}}}
plot the equation y = 4x^2 + (1200/x), where y = the surface area
{{{ graph( 300, 200, -6, 10, -100, 1000, 4x^2+(1200/x), 339) }}}
You can see minimum surface area occurs when x = 5.2 cm. Green: y=339
Find the minimum surface area
S.A. = 4(5.2^2) + {{{1200/5.2}}}
S.A. = 108.16 + 230.77
S.A = 338.93 sq/cm