# SOLUTION: An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of

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 Question 86483: An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of each side of the square to be cut out. a) Find the function V that represents the volume of the box in terms of x. Answer: b) Graph this function and show the graph over the valid range of the variable x.. Show Graph here. c) Using the graph, what is the value of x that will produce the maximum volume? Answer. Found 2 solutions by Nate, ankor@dixie-net.com:Answer by Nate(3500)   (Show Source): You can put this solution on YOUR website!An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of each side of the square to be cut out. a) Find the function V that represents the volume of the box in terms of x. Answer: Drawing a picture would be extremely helpful. 1. Draw a 4 by 6 foot rectangle 2. Make boxes at each of the four corners with sides x feet Length: 6 - 2x Width: 4 - 2x Height: x V = product of all sides V = (6 - 2x)(4 - 2x)(x) V = 4(3 - x)(2 - x)(x) V = 4(6 - 5x + x^2)(x) V = 4(6x - 5x^2 + x^3) V = 24x - 20x^2 + 4x^3 b) Graph this function and show the graph over the valid range of the variable x.. Show Graph here. c) Using the graph, what is the value of x that will produce the maximum volume? Answer. Approx. when x = 1 ~~~~ V(x) = 24x - 20x^2 + 4x^3 V'(x) = 24 - 40x + 12x^2 0 = 6 - 10x + 3x^2 ~ Exact: Answer by ankor@dixie-net.com(15746)   (Show Source): You can put this solution on YOUR website!An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of each side of the square to be cut out. : We know from the above information: Box length = (6-2x) Box width = (4-2x) Box height = x : a) Find the function V that represents the volume of the box in terms of x. Answer: Vol = length * width * height V = (6-2x)*(4-2x)*x V = x(24 - 20x + 4x^2); FOILed (6-2x)(4-2x); then mult by x V = 4x^3 - 20x^2 + 24x; represents the volume of the box : b) Graph this function and show the graph over the valid range of the variable x.. Plot the value from .2 to 1.8 only; y = V x | y ------- .2 | 4.032 .4 | 6.656 .6 | 8.064 .8 | 8.448 1.0| 8.0 1.2| 6.912 etc : Show Graph here. c) Using the graph, what is the value of x that will produce the maximum volume? Answer. : It looks like max volume will occur when x = .8 :