An open box is to be made from a rectangular sheet of metal that measures 10 by 8 inches, by cutting out squares of the same size from each corner and bending up the sides. Note The picture: A) Express the volume of the box as a function of x Simplify. B) Determine the domain of x f rom physical considerations. C) use the rational zero theorm and your answer from part b to find any rational value of x that will give the box a volume of 24 cubic inches. D) Find any value(s) of x that give(s) the box a desired volume of 24 cubic inches. An open box is to be made from a rectangular sheet of metal that measures 10 by 8 inches, by cutting out squares of the same size from each corner and bending up the sides. Note The picture: Express the volume of the box as a function of x Simplify. Determine the domain of x from physical considerations. use the rational zero theorem and your answer from part b to find any rational value of x that will give the box a volume of 24 cubic inches. Find any value(s) of x that give(s) the box a desired volume of 24 cubic inches. Solution A) V(x) = x(8-2x)(10-2x) = 4x(4-x)(5-x) B) x > 0 4-x > 0 -> x < 4 5-x > 0 -> x < 5 So the domain of x is: (0 , 4) C) V(x) = 24 -> 4x(4-x)(5-x) = 24 x(4-x)(5-x)= 6 -> x^3 - 9x^2 + 20x - 6 = 0 q|1 -> q = 1 p|(-6) -> p = 2 or 3 -> x = p/q -> x = 2 , -2 or x = 3 , -3 but from (b) : 0<x<4 -> x = 2 or 3 by testing the values: x = 3 D) x^3 - 9x^2 + 20x - 6 = (x-3)(x^2 - 6x + 2) = 0 -> roots are: x = 3 x = 3 - sqrt(7) x = 3 + sqrt(7) .