A sheet of paper 90 cm-by-66 cm is made into an open box (i.e. there's no top), by cutting x-cm squares out of each corner and folding up the sides. Find the value of x that maximizes the volume of the box. Give your answer in the simplified radical form. x =?? is the max.

Answer:[tex]26 - \sqrt{181}[/tex] cmStep-by-step explanation:The volume of the box is:V = height * length * widthV = x*(66 - 2*x)*(90 - 2*x) V = (66*x - 2*x^2)*(90 - 2*x) V = 5940*x - 132*x^2 - 180*x^2 + 4*x^3V = 4*x^3 - 312*x^2 + 5940*xwhere x is the length of the sides of the squares,  in cm.The mathematical problem is :Maximize: V = 4*x^3 - 312*x^2 + 5940*xsubject to: x > 02*x < 66 <=> x < 33In the maximum, the first derivative of V, dV/dx, is equal to zerodV/dx = 12*x^2 - 624*x + 5940From quadratic formula[tex]x = \frac{-b \pm \sqrt{b^2 - 4(a)(c)}}{2(a)} [/tex][tex]x = \frac{624 \pm \sqrt{(-624)^2 - 4(12)(5940)}}{2(12)} [/tex][tex]x = \frac{624 \pm \sqrt{104256}}{24} [/tex][tex]x = \frac{624 \pm \sqrt{2^6*3^2*181}}{24} [/tex][tex]x = \frac{624 \pm 8*3*\sqrt{181}}{24} [/tex][tex]x_1 = \frac{624 + 24*\sqrt{181}}{24} [/tex][tex]x_1 = 26 + \sqrt{181}[/tex][tex]x_2 = \frac{624 - 24*\sqrt{181}}{24} [/tex][tex]x_2 = 26 - \sqrt{181}[/tex]But [tex]x_1 > 33[/tex], then is not the correct answer.