A candy box is made from a piece of cardboard that measures 43 by 23 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume?
Accepted Solution
A:
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×HeightV = (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] = 0Now 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, 4.79Now 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.