Question

An open topped box is constructed as follows: A 24 in. x 24 in. square bottom has four identical squares cut off at each corner.

Answer 1

Explanation:

Given

square bottom area is
`24* 24\ in.^2`

Suppose x in. is cut from each corner to make a open box with maximum volume

New base area is
`(24-x)* (24-x) in.^2`

Volume of box

`V=(24-x)^2* x`

Differentiate V w.r.t x to get maximum volume

`\frac{\mathrm{d} V}{\mathrm{d} x}=2\cdot (24-x)(-1)+1\cdot (24-x)^2`

Put
`\frac{\mathrm{d} V}{\mathrm{d} x}=0`

`\left ( 24-x\right )\left [ -2x+24-x\right ]=0`

`\left ( 24-x\right )\left [ 24-3x\right ]=0`

`x=24,8`

but x=24 is not possible therefore x=8 will yield maximum volume

`V=(24-8)^2\cdot 8=2048\ in.^2`