Chapter 4.4, Problem 17Q

To determine

To find: the values of $x$ for which the model sense.

Answer to Problem 17Q

The model make sense for $0<x<1.5$ .

Explanation of Solution

Given information:

The volume $V$ of a rectangular wooden crate is modeled by the function $V\left(x\right)=2x^3-11x^2+12x$ , where $x$ is the width of the crate.

Given graph:

  

Calculation:

From the graph, the zero appears to be 0 and 4. The third zero is between 1 and 2.

Determine whether 0 is a zero using synthetic division.

  $\begin{array}{ccccc}0& 2& -11& 12& 0\\ & & 0& 0& 0\\ & 2& -11& 12& 0\end{array}$

Determine whether 4 is zero using synthetic division.

  $\begin{array}{cccc}4& 2& -11& 12\\ & & 8& -12\\ & 2& -3& 0\end{array}$

Since remainder is 0, so 4 is a zero or $x-4$ is a factor.

Write volume function as,

  $\begin{array}{c}V\left(x\right)=2x^3-11x^2+12x\\ =x\left(x-4\right)\left(2x-3\right)\end{array}$

Solve factor $2x-3=0$ for $x$ .

  $\begin{array}{c}2x=3\\ x=1.5\end{array}$

So, the zeros are 0,4 and 1.5.

Therefore, the model make sense for $0<x<1.5$ .