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](https://tex.z-dn.net/?f=area%20%5Ctimes%20lenth)
= ![a^2\times b](https://tex.z-dn.net/?f=a%5E2%5Ctimes%20b)
V = ![a^2\times b](https://tex.z-dn.net/?f=a%5E2%5Ctimes%20b)
puting the value of b
V = ![a^2 ( 108 - 4a )](https://tex.z-dn.net/?f=a%5E2%20%28%20108%20-%204a%20%29)
![V = 108a^2 - 4a^3](https://tex.z-dn.net/?f=V%20%3D%20108a%5E2%20-%204a%5E3)
To find the maximum value of V
(1) we differentiate it
![\frac{\mathrm{d} V}{\mathrm{d} a} = 216a - 12a^2](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cmathrm%7Bd%7D%20V%7D%7B%5Cmathrm%7Bd%7D%20a%7D%20%3D%20216a%20-%2012a%5E2)
(2) ![\frac{\mathrm{d} V}{\mathrm{d} a} = 0](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cmathrm%7Bd%7D%20V%7D%7B%5Cmathrm%7Bd%7D%20a%7D%20%20%3D%200)
= 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