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

Answer:

Step-by-step explanation:

Given

square bottom area is $24\times 24\ in.^2$

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

New base area is $(24-x)\times (24-x) in.^2$

Volume of box

$V=(24-x)^2\times x$

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

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