. 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

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 =

                         =  

    V  =  

puting the value of b

                      V  =

To find the maximum value of V  

(1)  we differentiate it

(2)  

= 0

12a ( 18 - a ) =

a = 0  and a = 18

(3)       putting the value of a if = negative then the value for a ,V  is maximum

      = 216 - 24a

put the value of a = 0 ,    = 216

put the value of a = 18  ,   negative

for the value of a =18  V gives maximum value

Max volume =    

                      =  11664

a = 18 ,  b = 108 - 4a = = 36