Chapter 4.7, Problem 27E

To determine

To find: the rule for $W$ and interpret the function $W\left(5\right)$ .

Answer to Problem 27E

  $W\left(5\right)=1105\text{yard}\text{.}$ .

Explanation of Solution

Given:

  $\begin{array}{l}V\left(x\right)=x^3-4x.\\ W\left(x\right)=V\left(3x\right).\end{array}$

Concept used:

A composite function is a function that depend on another function. A composite function is created when one function is substituted into another function.

for example $f\left(g\left(x\right)\right)$ is the composite function that is formed when $g\left(x\right)$ is substituted for $x$ in $f\left(x\right)$ .

The domain of $f$ becomes $g\left(x\right)$ .

In this case the value $g\left(x\right)$ is substitute in the domain of the $f\left(x\right)$ .

Calculation:

According to the given:

  $\begin{array}{l}V\left(x\right)=x^3-4x.\\ W\left(x\right)=V\left(3x\right).\end{array}$

By putting $3x$ in term of $x$ :

  $\begin{array}{l}W\left(x\right)=V\left(3x\right).\\ \Rightarrow {\left(3x\right)}^3-4\left(3x\right).\\ \Rightarrow 27x^3-12x.\\ W\left(5\right)=27{\left(5\right)}^3-12\left(5\right).\\ \Rightarrow 3315\text{ feet}\text{.}\end{array}$

Since, $1\text{ yard=}3\text{feet}$ .

  $W\left(5\right)=\frac{3315}{3}\text{yard}\text{.}$

Hence, $W\left(5\right)=1105\text{yard}\text{.}$ .