Chapter 4.4, Problem 55E

To determine

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

Answer to Problem 55E

The model make sense for $x>6.5$ .

Explanation of Solution

Given information:

The volume of shipping box is modeled by $V=2x^3-19x^2+39x$ , where $x$ is the length.

Given graph:

  

Calculation:

From the graph, the zero appears to be 0 and 3. The third zero is between 6 and 7.

Determine whether 0 is a zero using synthetic division.

  $\begin{array}{ccccc}0& 2& -19& 39& 0\\ & & 0& 0& 0\\ & 2& -19& 39& 0\end{array}$

Determine whether 3 is zero using synthetic division.

  $\begin{array}{cccc}3& 2& -19& 39\\ & & 6& -39\\ & 2& -13& 0\end{array}$

Since remainder is 0, so 3 is a zero or $x-3$ is a factor.

Write volume function as,

  $\begin{array}{c}V\left(x\right)=2x^3-19x^2+39x\\ =x\left(x-3\right)\left(2x-13\right)\end{array}$

Solve factor $2x-13=0$ for $x$ .

  $\begin{array}{c}2x=13\\ x=6.5\end{array}$

So, the zeros are 0,3 and 6.5.

Because dimension of box must be positive, so $x$ must be greater than 6.5.

Therefore, the model make sense for $x>6.5$ .