Question

Question #6d4ae

Answer 1

`2/3 " inch"`

Explanation:

Let the size of square to be cut from each corner `=x " inch"`
Volume of the open box so formed `V=lxxwxxh=(8-2 x)xx(3-2x)xx x`
`=>V=(24-22 x+4x^2)xx x`
`=>V=24x-22 x^2+4x^3`
To find maximum value we need to differentiate `V` with respect to `x` and equate it to `0`
`(dV)/dx=d/dx(24x-22 x^2+4x^3)`, setting it equal to zero we get
`24-44 x+12x^2=0`, dividing both sides by `4` and rearranging we get
`3x^2-11x+6=0`
Using split the middle term method to find the roots of above we get
`3x^2-9x-2x+6=0`
`=>3x(x-3)-2(x-3)=0`
`=>(x-3)(3x-2)=0`
We get the roots as
`x=3, 2/3`

For `x=3`, we see that width `w` of the box becomes negative. Hence, ignoring this root we have size of the square `2/3 " inch"`