Question

`a shipping company ships small crates forheavy materials that are secured by placingmetal strapping around each of the threedirections possible. each band must havean overlap of 2 inches for the crimpingdevice to tighten up the band, if the totallength of metal strapping is 222 inches, findthe dimensions(L, W, and H) of a box withmaximum volume the shipping companycan offer. also so find the maximumvolume.`

Answer 1

maximum volume = 5832 square inches

dimension for this max volume =

L = 18 inches

W = 18 inches

H = 18 inches

Answer 2

Trapping around each of 3 directions

$p_1 = 2(a+b)\\p_2 = 2(a+c)\\p_3 = 2(b+c)$        

which p is perimeter of each direction

Because each bond have an overlap 2 inches

$p_1_m =2(a+b)+2\\p_2_m = 2(a+c)+2\\p_3_m = 2(b+c)+2$

$p_1_m+p_2_m+p_3_m = 6+4(a+b+c)$

Total length of metal strapping is $222$ inches, we have

$6+4(a+b+c)\leq 222\\\Rightarrow 4(a+b+c)\leq 216\\\Rightarrow a+b+c \leq 54$

$V_b_0_x = a,b,c$   max =? when $a+b+c \leq 54$

$54 \geq a+b+c\geq 3^3\sqrt{abc}$

$V_b_o_x = abc \leq \frac{54}{3}^3 = 18^3$

maximum value V = $18^3 = 5832 (inch^3)$

at $a=b=c=18$ or

$L=W=H =18 (inches)$