Respuesta :
Answer:
The dimension of the box is 18.66 in by 2.66 in by 1.17 in.
Step-by-step explanation:
Given that, The cardboard is 21 in. long and 5 in. wide.
Assume,x be the length of the each sides of the square .
Then the length of the box is = (21-2x) in.
The breadth of the box is =(5-2x) in.
The height of the box is = length of side of the square
= x in.
The volume of the box = length × wide × height
=(21-2x)(5-2x)x
=(105-52x+4x²)x
=([tex]105 x-52x^2+4x^3[/tex])
Let
V=[tex]105 x-52x^2+4x^3[/tex]
Differentiating with respect to x
[tex]V'=105-104x+12x^2[/tex]
Again differentiating with respect to x
[tex]V''=-104+24x[/tex]
To find the dimension of the box, we set V'=0
[tex]105-104x+12x^2=0[/tex]
Applying the quadratic formula [tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex] , here a=12, b= -104 and c =105
[tex]\Rightarrow x=\frac{-(-104)\pm\sqrt{(-104)^2-4.12.105}}{2.12}[/tex]
[tex]=\frac{104\pm 76}{24}[/tex]
=7.5, 1.17
For x= 7.5 , the wide of the box will be negative.
∴x=1.17 in.
[tex]V''_{x=1.17}=-104+(24\times 1.17)<0[/tex]
Since at x= 1.17, V''<0 , therefore at x=1.17 in. the volume of the given box will be maximum.
The length of the box is = [21-(1.17×2)]=18.66 in.
The wide of the box is = [5-(1.17×2)]=2.66 in.
The height of the box is = 1.17 in.
The dimension of the box is 18.66 in by 2.66 in by 1.17 in.
The volume of a box is the amount of space in the box.
The dimensions that yield the maximum volume are: 18.60 by 2.60 by 1.20 inches
The dimension of the cardboard is:
[tex]\mathbf{Length = 21}[/tex]
[tex]\mathbf{Width = 5}[/tex]
Assume the cut-out is x.
So, the dimension of the box becomes
[tex]\mathbf{Length = 21 -2x}[/tex]
[tex]\mathbf{Width = 5 -2x}[/tex]
[tex]\mathbf{Height = x}[/tex]
The volume of the box is:
[tex]\mathbf{V = (21 - 2x) \times (5 - 2x) \times x}[/tex]
Open brackets
[tex]\mathbf{V = (21 - 2x) \times (5x - 2x^2)}[/tex]
Expand
[tex]\mathbf{V = 105x - 42x^2 -10x^2 + 4x^3}[/tex]
[tex]\mathbf{V = 105x -52x^2 + 4x^3}[/tex]
Differentiate
[tex]\mathbf{V' = 105 - 104x + 12x^2}[/tex]
Set to 0
[tex]\mathbf{105 - 104x + 12x^2 = 0}[/tex]
Rewrite as:
[tex]\mathbf{12x^2- 104x +105 = 0}[/tex]
Using a calculator, the value of x is:
[tex]\mathbf{x = (7.50, 1.20)}[/tex]
When x = 7.50;
[tex]\mathbf{Length = 21 -2(7.50) = 6}[/tex]
[tex]\mathbf{Width = 5 -2(7.50) = -10}[/tex]
[tex]\mathbf{Height = 7.50}[/tex]
The width cannot be negative.
When x = 1.20;
[tex]\mathbf{Length = 21 -2(1.20) = 18.60}[/tex]
[tex]\mathbf{Width = 5 -2(1.20) = 2.60}[/tex]
[tex]\mathbf{Height = 1.20}[/tex]
So, the dimensions that yield the maximum volume are: 18.60 by 2.60 by 1.20 inches
Read more about volumes at:
https://brainly.com/question/13529955
