Respuesta :
Answer:
For maximum volume of the box, squares with 4.79 inches should be cut off.
Step-by-step explanation:
A candy box is made from a piece of a cardboard that measures 43 × 23 inches.
Let squares of equal size will be cut out of each corner with the measure of x inches.
Therefore, measures of each side of the candy box will become
Length = (43 - 2x)
Width = (23 - 2x)
Height = x
Now we have to calculate the value of x for which volume of the box should be maximum.
Volume (V) = Length×Width×Height
V = (43 -2x)(23 - 2x)(x)
= [(43)×(23) - 46x - 86x + 4x²]x
= [989 - 132x + 4x²]x
= 4x³- 132x² + 989x
Now we find the derivative of V and equate it to 0
[tex]\frac{dV}{dx}=12x^{2}-264x+989[/tex] = 0
Now we get values of x by quadratic formula
[tex]x=\frac{264\pm \sqrt{264^{2}-4\times 12\times 989}}{2\times 12}[/tex]
x = 17.212, 4.79
Now we test it by second derivative test for the maximum volume.
[tex]\frac{d"V}{dx}= 24x - 264[/tex]
For x = 17.212
[tex]\frac{d"V}{dx}= 24(17.212)-264=413.088-264=149.088[/tex]
This value is > 0 so volume will be minimum.
For x = 4.79
[tex]\frac{d"V}{dx}=24(4.79)-264=-149.04[/tex]
-149.04 < 0, so volume of the box will be maximum.
Therefore, for x = 4.79 inches volume of the box will be maximum.
