Question

bisa has a piece of construction paper that she wants to use to make an open rectangular prism. she will cut a square with side length x from each corner of the paper, so the length and width is decreased by 2x as shown in the diagram. answer parts a to d. a. -∞<x<∞ b. x>4 c. 0<x<4 d. 4<x<5.5 c. what do the x - intercepts of the graph mean in this context? the intercepts 0, 4, and 5.5 represent the side lengths of the cut squares that will result in a box with zero volume. the intercept is not meaningful because it is not possible to cut this length from each corner of an 8 - inch side.

Explanation:

Step1: Consider the constraints on x

The length of the paper is 11 inches and the width is 8 inches. After cutting squares of side - length x from each corner, the length of the base of the open - rectangular prism is \(l = 11−2x\) and the width is \(w = 8 - 2x\). Since length and width must be non - negative, we also consider the practicality that we are cutting from an 8 - inch side. The maximum value of x is limited by the 8 - inch side. If we set \(8−2x=0\), we get \(x = 4\). Also, \(x>0\) because we are actually cutting something. So \(0 < x<4\).

Step2: Analyze x - intercepts

The volume of the open - rectangular prism \(V=(11 - 2x)(8 - 2x)x\). The x - intercepts of the graph of the volume function \(V(x)\) are the values of x for which \(V(x)=0\). Setting \(V(x)=0\), we have \((11 - 2x)(8 - 2x)x = 0\). Solving \(x=0\), \(11 - 2x=0\) (so \(x = 5.5\)) and \(8 - 2x=0\) (so \(x = 4\)). The value \(x = 0\) means no squares are cut (so no box is formed, volume is 0), \(x = 4\) makes the width of the base 0, and \(x = 5.5\) makes the length of the base 0. The value \(x = 5.5\) is not meaningful in the context of cutting from an 8 - inch side.

Answer:

c. \(0 < x<4\)
The intercepts \(0\), \(4\), and \(5.5\) represent the side lengths of the cut squares that will result in a box with zero volume. The intercept \(5.5\) is not meaningful because it is not possible to cut this length from each corner of an 8 - inch side.