Respuesta :
Answer:
x = 1.639 in
Step-by-step explanation:
With these instructions, the length of the box would be (14-2x), the width would be (8-2x), and the height would be x, so the equation for volume would be width*length*height, or
[tex]V=(14-2x)(8-2x)(x)[/tex]
[tex]V=4x^3-44x^2+112[/tex]
Now we solve for dV/dx because we want to find maximum volume
[tex]\frac{dV}{dx} =12x^2-88x+112[/tex]
[tex]\frac{dV}{dx} =4(3x^2-22x+28)[/tex]
Now we let dV/dx equal zero to find critical values
[tex]0=4(3x^2-22x+28)[/tex]
[tex]0=3x^2-22x+28[/tex]
[tex]x=\frac{11+\sqrt{37} }{3}[/tex], [tex]x=\frac{11-\sqrt{37} }{3}[/tex]
Now we see where dV/dx changes signs:
dV/dx changes from positive to negative at [tex]\frac{11-\sqrt{37} }{3}[/tex]
and from negative to positive at [tex]\frac{11+\sqrt{37} }{3}[/tex]
so V(x) has a local maximum when x = [tex]\frac{11-\sqrt{37} }{3} = 1.639[/tex] in
