Answer:
4,000 cm³
Step-by-step explanation:
Let x be the length of the sides of the base and h be the height of the box.
The surface area is given by:
![A=x^2+4xh=1,200\\h=\frac{1,200-x^2}{4x}](https://tex.z-dn.net/?f=A%3Dx%5E2%2B4xh%3D1%2C200%5C%5Ch%3D%5Cfrac%7B1%2C200-x%5E2%7D%7B4x%7D)
The volume if the box is:
![V=hx^2\\V=({\frac{1,200-x^2}{4x}} )*x^2\\V=300x-\frac{x^3}{4}](https://tex.z-dn.net/?f=V%3Dhx%5E2%5C%5CV%3D%28%7B%5Cfrac%7B1%2C200-x%5E2%7D%7B4x%7D%7D%20%29%2Ax%5E2%5C%5CV%3D300x-%5Cfrac%7Bx%5E3%7D%7B4%7D)
The value of x for which the derivate of the function above is zero will produce the largest possible volume:
![V=300x-\frac{x^3}{4} \\V'=300-\frac{3}{4}x^2=0\\ x^2=400\\x=20\ cm](https://tex.z-dn.net/?f=V%3D300x-%5Cfrac%7Bx%5E3%7D%7B4%7D%20%5C%5CV%27%3D300-%5Cfrac%7B3%7D%7B4%7Dx%5E2%3D0%5C%5C%20x%5E2%3D400%5C%5Cx%3D20%5C%20cm)
The height of the box is:
![h=\frac{1,200-20^2}{4*20}\\h=10\ cm](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B1%2C200-20%5E2%7D%7B4%2A20%7D%5C%5Ch%3D10%5C%20cm)
The largest possible volume is:
![V=10*20^2\\V=4,000\ cm^3](https://tex.z-dn.net/?f=V%3D10%2A20%5E2%5C%5CV%3D4%2C000%5C%20cm%5E3)
The largest possible volume of the box is 4,000 cm³