Chapter 4.4, Problem 56E

To determine

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

Answer to Problem 56E

The model make sense for $\left(1,\frac{5}{3}\right)$ .

Explanation of Solution

Given information:

The volume of a rectangular birdcage can be modeled by $V=2x^3-17x^2+29x-15$ , where $x$ is the length.

Given graph:

  

Calculation:

From the graph, the zero appears to be 1 and 3. The third zero is between 1 and 2.

Determine whether 1 is a zero using synthetic division.

  $\begin{array}{ccccc}1& 3& -17& 29& -15\\ & & 3& -14& 15\\ & 3& -14& 15& 0\end{array}$

Since remainder is 0, so 1 is a zero or $x-1$ is a factor.

Determine whether 3 is zero using synthetic division.

  $\begin{array}{cccc}3& 3& -14& 15\\ & & 9& -15\\ & 3& -5& 0\end{array}$

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

Write volume function as,

  $\begin{array}{c}V\left(x\right)=3x^3-17x^2+29x-15\\ =\left(x-1\right)\left(x-3\right)\left(3x-5\right)\end{array}$

Solve factor $3x-5=0$ for $x$ .

  $\begin{array}{c}3x=5\\ x=\frac{5}{3}\end{array}$

So, the zeros are 1,3 and $\frac{5}{3}$ .

Therefore, the model make sense for $\left(1,\frac{5}{3}\right)$ .