# Answer to Question #1425 in Calculus for Lauren

Question #1425

You are planning to make an open rectangular box from a 12 by 14 cm piece of cardboard by cutting congruent squares from the corners and folding up the sides.

<br>A) what are the dimensions of the box of largest volume you can make this way? <br>

B) what is the volume?

<br>A) what are the dimensions of the box of largest volume you can make this way? <br>

B) what is the volume?

Expert's answer

Denote the side of the square as x.

The volume of the box would be V = x*(12-2x)*(14-2x) = 4x

Let's find the roots of first derivation dV/dx:

V' = 12x2 - 104x + 168 = 0: x1 = 2 (13-2√37)=1.67 and x2 = 2(13 + 2√37)= 50.33 > 14

Thus the dimentions of the box would be:

{8.66, 10.66, 1.67}.

The volume of the box would be V = x*(12-2x)*(14-2x) = 4x

^{3}- 52x^{2 }+ 168xLet's find the roots of first derivation dV/dx:

V' = 12x2 - 104x + 168 = 0: x1 = 2 (13-2√37)=1.67 and x2 = 2(13 + 2√37)= 50.33 > 14

Thus the dimentions of the box would be:

{8.66, 10.66, 1.67}.

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