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A box is to be made out of a 10 by 14 piece of cardboard.

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A box is to be made out of a 10 by 14 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L , width  W, and height H  of the resulting box that maximizes the volume. (Assume that $$W \leq L$$ ).

L=

W=

H=

Jun 12, 2018

#1
+1

A box is to be made out of a 10 by 14 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L , width  W, and height H  of the resulting box that maximizes the volume. (Assume that  ).

L=

W=

H=

Hello Sloan!

$$V=(10-2x)*(14-2x)*x\\ =(140-20x-28x+4x^2)x\\ =4x^3-48x^2+140x$$

$$f(x)=4x^3-48x^2+140x\\ f'(x)=12x^2-96x+140=0$$

$$x = {{-b \pm \sqrt{b^2-4ac}} \over 2a}\\ x = {{96 \pm \sqrt{9216-4*12*140}} \over 2*12}\\ x=\frac{96\pm \sqrt{2496}}{24}=\frac{96\pm 49.95}{24}\\ x_1=6.082\ (inadmissible)\\ \color{blue}x_2=1.918$$

L = 14 - 2 * 1.918 = 10.163

W = 10 - 2 * 1.918 = 6,163

H = 1.918

Greetings !

Jun 12, 2018