Question

. The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 inches. What dimensions will give a box with a square end the largest volume

Answer 1

Answer:

Dimension a = 18 , b= 36 will give a box with a square end the largest volume

Step-by-step explanation:

Given -

sum of box length and girth (distance around) does not exceed 108 inches.

Let b be the lenth of box and a be the side of square

b  +  4a = 108

b = 108 - 4a                

Volume of box =$area \times lenth$

                         =  $a^2\times b$

    V  =  $a^2\times b$

puting the value of b

                      V  = $a^2 ( 108 - 4a )$

                     $V = 108a^2 - 4a^3$

To find the maximum value of V  

(1)  we differentiate it

 $\frac{\mathrm{d} V}{\mathrm{d} a} = 216a - 12a^2$

(2)   $\frac{\mathrm{d} V}{\mathrm{d} a} = 0$

$216a - 12a^2$ = 0

12a ( 18 - a ) =

a = 0  and a = 18

(3)       putting the value of a if $\frac{\mathrm{d^2} V}{\mathrm{d} a^2}$ = negative then the value for a ,V  is maximum

     $\frac{\mathrm{d^2} V}{\mathrm{d} a^2}$ = 216 - 24a

put the value of a = 0 ,  $\frac{\mathrm{d^2} V}{\mathrm{d} a^2}$  = 216

put the value of a = 18  ,  $\frac{\mathrm{d^2} V}{\mathrm{d} a^2} =$ negative

for the value of a =18  V gives maximum value

Max volume = $108\times18^2 - 4\times18^3$    

                      =  11664

a = 18 ,  b = 108 - 4a = $108 - 4\times 18$ = 36