Question

If 1900 square centimeters of material are available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

Answer:

Volume = $7969$ 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 $A^2 + 4*A*H.$

$A^2 + 4*A*H = 1900$

 Rearranging the above equation, we get -  

$`H = \frac{(1900 - A^2)}{(4*A)}$

Volume of box is equal to product of base area of box and the height of the box -  

$V = A*A* H$

Substituting the given area we get -

$\frac{A^2*(1900 - A^2)}{4A} = \frac{(1900*A - A^3)}{4}$

For maximum volume

$\frac{dV}{dA} =0$

$\frac{ 1900}{4} - \frac{3*A^2}{4} = 0$

$A^2 = \frac{1900}{3}$

Volume of the box

= $\frac{\frac{1900}{3}*(1900 - \frac{1900}{3}) }{4 * \sqrt{\frac{1900}{3} } }$

= $7969$ cubic centimeter