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. what is a reasonable domain for the function? a. x > 4 b. -∞ < x < ∞ c. 4 < x < 5.5 d. 0 < x < 4 what do the x - intercepts of the graph mean in this context? the x - intercepts represent the side lengths of the cut squares that will result in a box with volume. the x - intercept is not meaningful because it is not possible to cut this length from each corner of an 8 - inch side.

Explanation:

Step1: Analyze the physical constraints

The length of the paper is 11 inches and width is 8 inches. We are cutting squares of side - length \(x\) from each corner to make an open - top rectangular prism. The side - length \(x\) must be non - negative (\(x\geq0\)). Also, we can't cut more than half of the shorter side of the paper. Since the shorter side of the paper is 8 inches, if we cut \(x\) from both ends of the shorter side, \(2x\leq8\), so \(x\leq4\).

Step2: Determine the domain

Combining the non - negativity condition \(x\geq0\) and the upper - bound condition \(x\leq4\), the domain of the function (where the function likely represents the volume of the box) is \(0 < x<4\). The \(x\) - intercepts of a graph in this context represent the values of \(x\) for which the volume of the box is zero. If \(x = 0\), we haven't cut any squares and the box has no height, so the volume is zero. If \(x = 4\), we have cut away so much that the width of the base of the box becomes zero (since \(8-2x=0\) when \(x = 4\)), and the volume is also zero. The \(y\) - intercept is not meaningful because \(x\) (the side - length of the cut - out square) cannot be negative, and the \(y\) - intercept would occur when \(x = 0\) which represents a non - box (a flat piece of paper).

Answer:

The reasonable domain for the function is \(0 < x<4\). The \(x\) - intercepts represent the values of \(x\) for which the volume of the open - top rectangular prism is zero (either no cut is made (\(x = 0\)) or the cut is so large that the base of the box has zero width (\(x = 4\))). The \(y\) - intercept is not meaningful because negative values of \(x\) (which would be related to the \(y\) - intercept in a coordinate sense) are not possible in the context of cutting squares from the paper.