Problem 93 - College Algebra

Question

The volume, \(V,\) of a rectangular solid with length \(l,\) width \(w,\) and height \(h\) is given by the formula \(V\) - lwh. Use this formula to write a polynomial in standard form that models, or represents, the volume of the open box. (Object can copy)

Step-by-Step Solution

Answer

The volume of the open box can be modeled by the polynomial \(V = 10x^3\).

Step 1: Understand the Problem

The volume of a rectangular solid is given by the formula \(V = lwh\). Our task is to rewrite this as a polynomial. A polynomial in standard form is written as \(ax^n + bx^{n-1} + ... + c\), where \(a, b, c\) are coefficients, \(x\) is the variable, and \(n\) is the degree of the polynomial.

Step 2: Rewrite the Formula

Unfortunately, the volume formula does not include any added or subtracted terms so it can't be directly rewritten as a polynomial. However, if we allow for \(l, w,\) and \(h\) to be functions of a variable \(x,\) we can, for example, represent the volume as a polynomial in \(x\). Let's represent \(l\) as \(x\), \(w\) as \(5x\), and \(h\) as \(2x\). Then the volume \(V\) becomes \(V = x * 5x * 2x = 10x^3\).

Step 3: Result

The result is a polynomial that represents the volume of the open box. In this case it is \(V = 10x^3\).