Respuesta :
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 =[tex]area \times lenth[/tex]
= [tex]a^2\times b[/tex]
V = [tex]a^2\times b[/tex]
puting the value of b
V = [tex]a^2 ( 108 - 4a )[/tex]
[tex]V = 108a^2 - 4a^3[/tex]
To find the maximum value of V
(1) we differentiate it
[tex]\frac{\mathrm{d} V}{\mathrm{d} a} = 216a - 12a^2[/tex]
(2) [tex]\frac{\mathrm{d} V}{\mathrm{d} a} = 0[/tex]
[tex]216a - 12a^2[/tex] = 0
12a ( 18 - a ) =
a = 0 and a = 18
(3) putting the value of a if [tex]\frac{\mathrm{d^2} V}{\mathrm{d} a^2}[/tex] = negative then the value for a ,V is maximum
[tex]\frac{\mathrm{d^2} V}{\mathrm{d} a^2}[/tex] = 216 - 24a
put the value of a = 0 , [tex]\frac{\mathrm{d^2} V}{\mathrm{d} a^2}[/tex] = 216
put the value of a = 18 , [tex]\frac{\mathrm{d^2} V}{\mathrm{d} a^2} =[/tex] negative
for the value of a =18 V gives maximum value
Max volume = [tex]108\times18^2 - 4\times18^3[/tex]
= 11664
a = 18 , b = 108 - 4a = [tex]108 - 4\times 18[/tex] = 36
