Respuesta :
Answer:
Volume = [tex]7969[/tex] cubic centimeter
Step-by-step explanation:
Let the length of each side of the base of the box be A and the height of the box be H.
Area of material required to make the box is equal to is [tex]A^2 + 4*A*H.[/tex]
[tex]A^2 + 4*A*H = 1900[/tex]
Rearranging the above equation, we get -
[tex]`H = \frac{(1900 - A^2)}{(4*A)}[/tex]
Volume of box is equal to product of base area of box and the height of the box -
[tex]V = A*A* H[/tex]
Substituting the given area we get -
[tex]\frac{A^2*(1900 - A^2)}{4A} = \frac{(1900*A - A^3)}{4}[/tex]
For maximum volume
[tex]\frac{dV}{dA} =0[/tex]
[tex]\frac{ 1900}{4} - \frac{3*A^2}{4} = 0[/tex]
[tex]A^2 = \frac{1900}{3}[/tex]
Volume of the box
= [tex]\frac{\frac{1900}{3}*(1900 - \frac{1900}{3}) }{4 * \sqrt{\frac{1900}{3} } }[/tex]
= [tex]7969[/tex] cubic centimeter
