Chapter 1.8, Problem 64E

a .

To determine

To write: function R in terms of p giving cost of hybrid car after receiving the rebate from the factory

Answer to Problem 64E

  $R\left(p\right)=p-2000$

Explanation of Solution

Given:

The retail price of a new hybrid car is p dollars. The dealership advertises a factory rebate of $2000.

Calculation:

The function R is obtained by subtracting the rebate from the price of the car. So,

  $R\left(p\right)=p-2000$

b.

To determine

To write: function S in terms of p giving cost of hybrid car after receiving the dealership discount.

Answer to Problem 64E

  $S\left(p\right)=0.9p$

Explanation of Solution

Given:

The retail price of a new hybrid car is p dollars. The dealership advertises a 10% discount.

Calculation:

The function R is obtained by subtracting the 10% of price of car from the price of the car. So,

  $\begin{array}{c}S\left(p\right)=p-\frac{10}{100}p\\ =p-0.1p\\ =0.9p\end{array}$

c.

To determine

To find: and interpret $\left(R\circ S\right)\left(p\right)$ and $\left(S\circ R\right)\left(p\right)$ .

Answer to Problem 64E

  $\left(R\circ S\right)\left(p\right)=0.9p-2000$

It represents the price of car when dealership discount is applied before the factory rebate.

  $\left(S\circ R\right)\left(p\right)=0.9p-1800$

It represents the price of car when factory rebate is applied before the dealership discount.

Explanation of Solution

Calculation:

From part (a), $R\left(p\right)=p-2000$

From part (b), $S\left(p\right)=0.9p$

Consider,

  $\begin{array}{c}\left(R\circ S\right)\left(p\right)=R\left(S\left(p\right)\right)\\ =R\left(0.9p\right)\\ =0.9p-2000\end{array}$

It represents the price of car when dealership discount is applied before the factory rebate.

And,

  $\begin{array}{c}\left(S\circ R\right)\left(p\right)=S\left(R\left(p\right)\right)\\ =S\left(p-2000\right)\\ =0.9\left(p-2000\right)\\ =0.9p-1800\end{array}$

It represents the price of car when factory rebate is applied before the dealership discount.

d.

To determine

To find: $\left(R\circ S\right)\left(25,795\right)$ and $\left(S\circ R\right)\left(25,795\right)$ . And determine which yields the lower cost for the hybrid car

Answer to Problem 64E

  $\left(R\circ S\right)\left(25,795\right)=\$21215.50$

  $\left(S\circ R\right)\left(25,795\right)=\$21415.50$

  $\left(R\circ S\right)\left(25,795\right)$ yields a lower cost.

Explanation of Solution

Calculation:

From part (c), $\left(R\circ S\right)\left(p\right)=0.9p-2000$ and $\left(S\circ R\right)\left(p\right)=0.9p-1800$

Consider,

  $\begin{array}{c}\left(R\circ S\right)\left(25,795\right)=0.9\left(25,795\right)-2000\\ =23215.5-2000\\ =\$21215.50\end{array}$

And,

  $\begin{array}{c}\left(S\circ R\right)\left(25,795\right)=0.9\left(25,795\right)-1800\\ =23215.5-2000\\ =\$21415.50\end{array}$

Clearly, $\left(R\circ S\right)\left(25,795\right)$ yields a lower cost. This implies that dealership discount should be applied before the factory rebate.