Answer:
The dimension of the box is 17.66 in by 7.66 in by 2.67 in.
Therefore the volume of the box is 361.19
.
Step-by-step explanation:
Given that the dimensions of a cardboard is 23 in by 13 in.
Let the side of the square be x in.
Then the length of the box= (23-2x) in
and the width of the box =(13-2x) in
and height = x in.
The volume of the box is = length ×width × height
=[(23-2x)(13-2x)x] ![in^3](https://tex.z-dn.net/?f=in%5E3)
=(299x-72x² +4
) ![in^3](https://tex.z-dn.net/?f=in%5E3)
∴V=299x-72x² +4x³
Differentiating with respect x
V'= 299-144x+12x²
Again differentiating with respect x
V''= -144+24x
To find the dimensions, we set V'=0
∴299-144x+12x²=0
Applying Sridharacharya formula that is the solution of a quadratic equation ax²+bx+c is
Here a=12, b=-144 , c=299
![\therefore x=\frac{-(-144)\pm\sqrt{(-144)^2-4.12.299}}{2.12}](https://tex.z-dn.net/?f=%5Ctherefore%20x%3D%5Cfrac%7B-%28-144%29%5Cpm%5Csqrt%7B%28-144%29%5E2-4.12.299%7D%7D%7B2.12%7D)
![\Rightarrow x= 9.33, 2.67](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D%209.33%2C%202.67)
If we take x=9.33 in, then the width of the box [13-(2×9.33)] will negative.
∴x = 2.67 in
If at x = 2.67, V''<0 , then the volume of the box will be maximum or V''>0 then volume of the box will be minimum.
![V''|_{x=2.67}=-144+(24\times 2.67)=-79.92](https://tex.z-dn.net/?f=V%27%27%7C_%7Bx%3D2.67%7D%3D-144%2B%2824%5Ctimes%202.67%29%3D-79.92%3C0)
Therefore at x = 2.67, the volume of the box maximum.
The length of the box =[23-(2×2.67)] in
=17.66 in
The width of the box =[13-(2×2.67)] in
=7.66 in
The height of the box= 2.67 in
The dimension of the box is 17.66 in by 7.66 in by 2.67 in.
Therefore the volume of the box is =(17.66×7.66×2.67) ![in^3](https://tex.z-dn.net/?f=in%5E3)
=361.19 ![in^3](https://tex.z-dn.net/?f=in%5E3)