Respuesta :
Answer:
Therefore the dimensions of the box is 10.82 inches × 6.82 inches ×2.09 inches.
Step-by-step explanation:
Given that, by cutting equal squares from each corner form a rectangular sheet of material, an open top box is constructed.
The dimensions of the material is 15 inches by 11 inches.
Assume that,the side of the square be x.
The length of the box is= (15 -2x) inches
The width of the box is =(11-2x) inches
The height of the box is = x inches.
The volume of the box is = Length× width×height
=(15-2x)(11-2x)x cubic inches
=(165-52x+4x²)x cubic inches
[tex]=165x -52x^2+4x^3[/tex] cubic inches
Let,
V[tex]=165x -52x^2+4x^3[/tex]
Differentiating with respect to x
V'= [tex]165-104x+12x^2[/tex]
Again differentiating with respect to x
V''= -104 +24x
Now, we set V'=0
[tex]165-104x+12x^2=0[/tex]
[tex]\Rightarrow x=\frac{-(-104)\pm\sqrt{(-104)^2-4.12.165}}{2.12}[/tex]
[tex]\Rightarrow x=\frac{104\pm \sqrt{2896}}{24}[/tex]
[tex]\Rightarrow x=\frac{104\pm 4\sqrt{181}}{24}[/tex]
[tex]\Rightarrow x= 2.09,6.58[/tex]
For x=6.58 the width of the box will be negative.
∴x=2.09
[tex]V''|_{x=2.09}=-104+(24\times 2.09)=-5384<0[/tex]
Since V''<0 at x= 2.09,
Therefore V is maximum when x=2.09.
The length of the box is = {15-(2×2.09)}=10.82 inches
The width of the box is ={11-(2×2.09)}=6.82 inches
The height of the box is 2.09 inches.
Therefore the dimensions of the box is 10.82 inches × 6.82 inches ×2.09 inches.
