Answer:
2.67 inches.
Step-by-step explanation:
Assuming that we represent the size of the squares with the letter y, such that after the squares are being cut from each corner, the rectangular length of the box that is formed can now be ( 23 - 2y), the width to be (13 - 2y) and the height be (x).
The formula for a rectangular box = L × B × W
= (23 -2y)(13-2y) (y)
= (299 - 46y - 26y + 4y²)y
= 299y - 72y² + 4y³
Now for the maximum volume:
dV/dy = 0
This implies that:
299y - 72y² + 4y³ = 299 - 144y + 12y² = 0
By using the quadratic formula; we have :
[tex]= \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}[/tex]
where;
a = 12; b = -144 and c = 299
[tex]= \dfrac{-(-144) \pm \sqrt{(-144)^2 -4(12)(299)}}{2(12)}[/tex]
[tex]= \dfrac{144 \pm \sqrt{20736 -14352}}{24}[/tex]
[tex]= \dfrac{144 \pm \sqrt{6384}}{24}[/tex]
[tex]= \dfrac{144 \pm79.90}{24}[/tex]
[tex]= \dfrac{144 + 79.90}{24} \ \ OR \ \ \dfrac{144 - 79.90}{24}[/tex]
[tex]= 9.33 \ \ OR \ \ 2.67[/tex]
Since the width is 13 inches., it can't be possible for the size of the square to be cut to be 9.33
Thus, the size of the square to be cut out from each corner to obtain the maximum volume is 2.67 inches.
