Chapter 6.1, Problem 37PPS

(a).

To determine

Write two functions representing the price after the discount p(x) and the price after sales tax t(x).

Answer to Problem 37PPS

  $p(x)=x\cdot \left(1-\frac{35}{100}\right)=0.65x$

  $t(x)=x\cdot \left(1+\frac{6.25}{100}\right)=1.0625x$

Explanation of Solution

Given:

Ms.Smith wants to buy an HDTV , which is on sale for 35% off the original price of $2299.The sales tax is 6.25%.

Calculation:

Let the original price of the HDTV = x

Discount = 35%

So, the price after discount = original price − discount = $x-35\%x=x\cdot \left(1-\frac{35}{100}\right)=\frac{65}{100}x=0.65x$

So, the function for the price after discount :

  $p(x)=x\cdot \left(1-\frac{35}{100}\right)=0.65x$

Sales tax rate = 6.25%

So, the price after sales tax is added to it = original price + sales tax = $x+6.25\%x=x\cdot \left(1+\frac{6.25}{100}\right)=\frac{106.25}{100}x=1.0625x$

So, the function for the price after sales tax :

  $t(x)=x\cdot \left(1+\frac{6.25}{100}\right)=1.0625x$

(b).

To determine

Find the composition of functions that represents the price of the HDTV , $\left[p\circ t\right](x)\text{ or }\left[t\circ p\right](x)$ .

Answer to Problem 37PPS

  $\left[t\circ p\right](x)$ represents the price of the HDTV.

Explanation of Solution

Given:

Ms.Smith wants to buy an HDTV , which is on sale for 35% off the original price of $2299.The sales tax is 6.25%.

Calculation:

From part (a) , we have:

The function for the price after discount :

  $p(x)=x\cdot \left(1-\frac{35}{100}\right)=0.65x$

The function for the price after sales tax :

  $t(x)=x\cdot \left(1+\frac{6.25}{100}\right)=1.0625x$

Since, Ms.Smith pays sales tax on the discounted price of the HDTV, we need to first apply the function that gives the price of HDTV after the discount of 35% and then apply the function that applies the sales tax on the discounted price of the HDTV.

Hence, the correct composition is :

  $\left[t\circ p\right](x)=t\left[p(x)\right]$ since it gives the output of price after the sales tax on the discounted price of HDTV (input function of t(x)).

So,

  $\left[t\circ p\right](x)$ represents the price of the HDTV.

(c).

To determine

Find the price that Ms. Smith will pay for the HDTV.

Answer to Problem 37PPS

The price of HDTV after the discount and the sales tax is approximately $1587.75

Explanation of Solution

Given:

Ms.Smith wants to buy an HDTV , which is on sale for 35% off the original price of $2299.The sales tax is 6.25%.

Calculation:

From part (a) , we have:

The function for the price after discount :

  $p(x)=x\cdot \left(1-\frac{35}{100}\right)=0.65x$

The function for the price after sales tax :

  $t(x)=x\cdot \left(1+\frac{6.25}{100}\right)=1.0625x$

and from part (b) , we have:

  $\left[t\circ p\right](x)$ represents the price of the HDTV.

We have , x = $2299

  $\begin{array}{l}\left[t\circ p\right](x)\\ =t\left[p(x)\right]\\ =t\left(x\cdot \left(1-\frac{35}{100}\right)\right)\mathrm{......................}[\text{calculate 35\% discount on original price of HDTV]}\\ =t\left(2299\cdot \left(\frac{65}{100}\right)\right)\mathrm{.....................}[\text{original price of HDTV=\$2299]}\\ =t\left(1494.35\right)\\ =1494.35\cdot \left(1+\frac{6.25}{100}\right)\mathrm{...............}[\text{calculate 6}\text{.25\% sales tax on the discounted price of HDTV]}\\ =1494.35\cdot \left(\frac{106.25}{100}\right)\\ =1587.746875\\ \approx 1587.75\end{array}$

So, the price of HDTV after the discount and the sales tax is approximately $1587.75