A piece of cardboard is 2.4 times as long as it is wide. It is to be made into a

Question

A piece of cardboard is 2.4 times as long as it is wide. It is to be made into a box with an open top by cutting 3-inch squares from each comer and folding up the sides. Let x represent the width (in inches) of the original piece of cardboard Answer the following questions 3 a) Represent the length of the original piece of cardboard in terms of x. Length 2.4x in. (Use integers or decimals for any numbers in the expression.) b) Give the restrictions on x. What will be the dimensions of the bottom rectangular base of the box? The restriction on x will be x>6. (Type an inequality.) The length will be (2.4x-6)in. and the width will be (x-6)in (Type expressions using x as the variable. Use integers or decimals for any numbers in the expressions.) c) Determine a function V that represents the volume of the box in terms of x. V 7.2x61.2x108in (Simplify your answer. Use integers or decimals for any numbers in the expression.) d) For what dimensions of the bottom of the box will the volume be 520 in.3? The length will be 25.03] in and the width will be [693] in. Round to the nearest tenth as needed.) e) Find the values of x if such a box is to have a volume between 600 and 800 in.3 Between which two values must x be in order to produce this range of volumes? (Use a comma to separate answers as needed. Round to the nearest tenth as needed.)

Explanation / Answer

let width = x

length = 2.4 x

b) restriction is x > 6

length = 2.4x - 6

width = x - 6

c) volume = length * width * height

= ( 2.4x - 6) (x - 6) ( 3 )

=7.2x^2 - 61.2x + 108

e) 600< 7.2x^2 - 61.2x + 108 < 800

solving the inequality we get

13.5 < x < 14.9

hence, value of x must lie between

13.5 , 14.9

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