Answer:
Therefore, the length of the square is
cm which should be cut out of each corner to get a box with the maximum volume.
Step-by-step explanation:
Given that, an open box is made from a plastic with dimension 70 cm by 96 cm by removing a square from each corner of the plastic.
Consider, x be the length of the side of the square.
After cutting the corner, the length of the plastic is = (70-2x) cm
The width of the plastic is=(96-2x) cm.
Now the length of the open box = (70-2x) cm
The width of the open box = (96-2x) cm.
Then x will be the height of the open box.
The volume of the box is
V=Length×width×height
=[(70-2x) (96-2x) x] cm³
=(6720x -192x²-140x²+4x³) cm³
=(4x³-332x²+6720x) cm³
∴V=4x³-332x²+6720x
Differentiating with respect to x
V'= 12x²-664x+6720
Again differentiating with respect to x
V''= 24x -664
To find maximum volume, we set V' =0
12x²-664x+6720=0
⇒4(3x²-166x+1680)=0
⇒3x²-166x+1680=0
⇒3x²-126x-40x+1680=0
⇒3x(x-42)-40(x-42)=0
⇒(x-42)(3x-40)=0
![\Rightarrow x = 42,\frac{40}{3}](https://tex.z-dn.net/?f=%5CRightarrow%20x%20%3D%2042%2C%5Cfrac%7B40%7D%7B3%7D)
Now
![V''|_{x=42}=(24\times 42)-664=334>0](https://tex.z-dn.net/?f=V%27%27%7C_%7Bx%3D42%7D%3D%2824%5Ctimes%2042%29-664%3D334%3E0)
And
At
, V has maximum value.
Therefore, the length of the square is
cm which should be cut out of each corner to get a box with the maximum volume.