Quadratic equation

Question

A rectangular piece of cardboard measuring 12 ft by 10 ft is to be made into a box by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches. (a) (6 points) Give the restrictions on x. < x <

(b) (3 points) Determine a function V that gives the volume of the box as a function of x.

(c) (3 points) What is the name of this type of model?

(d) (3 points) Sketch the graph of the model.

Answer 1

0<x<5

V=(12-2x)(10-2x)x= (120-44x+ 4x²)x = 120x-44x^2+4x^3

take the derivative, set = 0 solve for x

V' =12x^2-88x+120 = 0

3x^2-22x+30 = 0

x = 22/6 + or - (1/6)sqr(22^2 - 4(3)(30))

= 11/3 + or - (1/6)sqr(484-360)

= 3 2/3 + or - (1/6)sqr(124)

= 3.667 + or - 1.856

= 1.811, 5.223, but x<5 because x=5 makes V=0, 5<x<6 makes V<0, so

x= 1.811 inches maximizes volume

V=(12-3.622)(10-3.622)(1.811) = (8.378)(6.378)(1.811)= 96.757 cubic inches

V'(x) is graphically a parabola or quadratic

V(x) is a cubic function with potential zeroes, or x-intercepts where V=0, at x=5, x=6 and x=0

since volume = (12-2x)(10-2x)x. Just set each factor = 0 and solve for x, getting 6,5 & 0.

It has a local max, local min and extends to infinity and negative infinity. A local min is between 5 and 6, a local max between 0 and 5.