Respuesta :
Answer:
a) The volume of the largest box that can be formed:[tex]567.42 ft^3[/tex]
b) The maximum volume of the box:[tex]\frac{2s^3}{27}[/tex]
Explanation:
a) Length of the card board = l = 27 ft
Width of the card board= b = 15 ft
Squares with sides of length x are cut out of each corner of a rectangular cardboard to form a box.
Now, length of the box = L = 27 -2x
Breadth of the box ,B= 15 - 2x
Height of the box ,H= x
Volume of the box ,V= L × B × H
[tex]\frac{dV}{dx}=\frac{(27 -2x)(15-2x)x}{dx}[/tex]
[tex]\frac{dV}{dx}=\frac{d(405x-84x^2+4x^3)}{dx}[/tex]
[tex]\frac{dV}{dx}=405-168x+12x^2[/tex]
Putting ,[tex]\frac{dV}{dx}=o[/tex]
[tex]0=405-168x+12x^2[/tex]
x = 10.9051, 3.0949
[tex]\frac{d^2V}{(dx)^2}=-168+24x[/tex]
When , x = 3.095 , [tex]\frac{d^2V}{(dx)^2}<0[/tex] (maxima)
The volume of the largest box that can be formed:
[tex](27 -2x)(15-2x)x = (27 -2\times 3.095)(15-2\times 3.095)\times 3.095[/tex]
[tex]V=567.42 ft^3[/tex]
b) Original piece of cardboard is a square with sides of length s.
Length of the card board = l = s
Squares with sides of length x are cut out of each corner of a rectangular cardboard to form a box.
Now, length of the box = L = s -2x
Height of the box ,H = x
Volume of the box ,V= L × L × H
[tex]\frac{dV}{dx}=\frac{(s -2x)(s-2x)x}{dx}[/tex]
[tex]\frac{dV}{dx}=\frac{d((s-2x)^2x)}{dx}[/tex]
[tex]\frac{dV}{dx}=(s-2x)^2+x(2(s-2x)(-2))[/tex]
[tex]\frac{dV}{dx}=(s-2x)^2-8x(s-2x)[/tex]
[tex]\frac{dV}{dx}=O[/tex]
[tex]x=\frac{s}{6},\frac{s}{2}[/tex]
[tex]\frac{d^2V}{(dx)^2}=2(s-2x)(-2)-8(2-2x)+16x[/tex]
When , [tex]x=\frac{s}{6}[/tex] , [tex]\frac{d^2V}{(dx)^2}<0[/tex] (maxima)
The maximum volume of the box:
[tex]V_{max}=(s-2x)^2x=(s-2\times \frac{s}{6})^2\times \frac{s}{6}=\frac{2s^3}{27}[/tex]
