Question

elena is making an open - top box by cutting squares out of the corners of a piece of paper that is 11 inches wide and 17 inches long, and then folding up the sides. if the side lengths of her square cutouts are x inches, then the volume of the box is given by ( v(x)=x(11 - 2x)(17 - 2x) ).
elena graphs the volume of the box along with the function ( b(x)=140 ).
a. what is a reasonable domain for ( v(x) )?
( <x< )
b. approximately which value of x will give her a box with the greatest volume?
c. for approximately which values of x is the volume of the box increasing?
d. what do the points of intersection of these two graphs represent?

Explanation:

Step1: Determine the domain of \(V(x)\)

Since \(x>0\) (side - length of the square) and \(11 - 2x>0\) (width of the base of the box) and \(17 - 2x>0\) (length of the base of the box).
Solving \(11 - 2x>0\) gives \(x < 5.5\), solving \(17 - 2x>0\) gives \(x<8.5\). The most restrictive inequality is \(x < 5.5\). So the domain is \(0

Step2: Find the value of \(x\) for maximum volume

Looking at the graph of \(y = V(x)\), the vertex (maximum point) occurs at \(x\approx2\) (by visual inspection of the graph).

Step3: Determine where \(V(x)\) is increasing

A function \(y = f(x)\) is increasing when the slope of the tangent line is positive. Looking at the graph of \(y = V(x)\), it is increasing on the interval \(0

Step4: Interpret the intersection points

The points of intersection of \(y = V(x)\) and \(y = B(x)=140\) represent the values of \(x\) for which the volume of the box \(V(x)\) is equal to \(140\) cubic inches.

Answer:

a. \(0 < x<5.5\)
b. \(x\approx2\)
c. \(0 < x<2\)
d. The values of \(x\) for which the volume of the box is \(140\) cubic inches.