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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.

Sagot :

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

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