91Ó°ÊÓ

You want to make an open box from a rectangular piece of material, 15 centimetres by 9 centimetres, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

Modeling Polynomials Sketch the graph of a polynomial function that is of fourth degree, has a zero of multiplicity \(2,\) and has a negative leading coefficient. Sketch the graph of another polynomial function with the same characteristics except that the leading coefficient is positive.

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive and that has a zero at \(x=3\) of multiplicity 2.