Consider the following problem: A box with an open top is to be constructed from
ID: 3188053 • Letter: C
Question
Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes.
(b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
(c) Write an expression for the volume V in terms of x and y.
V =
(d) Use the given information to write an equation that relates the variables x and y.
(e) Use part (d) to write the volume as a function of x.
V(x) =
Explanation / Answer
a) + b) You are on your own. c) y = length y = width x = height V = (y^2)x d) 3' = side of board = x + y + x = 2x + y e) 3 = 2x + y 2x + y =3 y = 3 - 2x V(x) = (y^2)x = [(3 - 2x)^2]x = [9 - 12x + 4(x^2)]x f) V'(x) = 0 = [9 - 12x + 4(x^2)] + x(-12 + 8x) 0 = 9 - 12x + 4(x^2) - 12x + 8(x^2) = 9 - 24x + 12(x^2) = 3 - 8x + 4(x^2) Use the quadratic equation and solve. x = [8 +/- sqrt(64 - 48)]/8 = [8 +/- sqrt(16)]/8 x = [8 +/- 4]/8 = (2 +/- 1)/2 = {3/2,1/2} at x = 1.5' base is 0' X 0' = 0 sq ft height = 1.5' Volume = 0 cubic feet: This is a minimum. at x = 1/2' base is 2' X 2' = 4 sq ft height = 0.5' Volume = 2 cubic feet Checking the ans: dx= 0.1' at x = 0.6', base is (3' - 1.2')^2 = 3.24 sq ft V = 1.944 cubic ft. (less as expected) at x = 0.4', base is 2.2^2 = 4.84 sq ft V = 1.936 cu ft. (less as expected)