Chapter 4.8, Problem 55PPS

(a)

To determine

To Draw: The graph of the inequality for monthly profit.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Function used by manufacturer to model monthly profit is }P\left(x\right)=x\left(-27.5x+3520\right)+20000\\ \text{Where }x\text{ is the no}\text{. of digital audio players}\\ \text{Monthly profit is at least }=\$100000\end{array}$

Formula Used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $\begin{array}{l}\text{Function used by manufacturer to model monthly profit is }P\left(x\right)=x\left(-27.5x+3520\right)+20000\\ \text{Where }x\text{ is the no}\text{. of digital audio players}\\ \text{Monthly profit is at least }=\$100000\\ P\left(x\right)\ge 100000\\ P\left(x\right)=x\left(-27.5x+3520\right)+20000\ge 100000\\ \Rightarrow x\left(-27.5x+3520\right)+20000-100000\ge 0\\ \Rightarrow -27.5x^2+3520x-80000\ge 0\\ \Rightarrow x=\frac{-3250\pm \sqrt{{3520}^2-4\left(-27.5\right)\left(-80000\right)}}{2\left(-27.5\right)}\\ \Rightarrow x=\frac{-3250\pm \sqrt{12390400-8800000}}{-55}\\ \Rightarrow x=\frac{3250+\sqrt{3590400}}{55}or\frac{3250-\sqrt{3590400}}{55}\\ \Rightarrow x=29.55or98.45\end{array}$

  

(b)

To determine

To Find: The total number of audio player sold.

Answer to Problem 55PPS

  $29550or98450$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Function used by manufacturer to model monthly profit is }P\left(x\right)=x\left(-27.5x+3520\right)+20000\\ \text{Where }x\text{ is the no}\text{. of digital audio players}\\ \text{Monthly profit is at least }=\$100000\end{array}$

Formula Used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $\begin{array}{l}\text{Function used by manufacturer to model monthly profit is }P\left(x\right)=x\left(-27.5x+3520\right)+20000\\ \text{Where }x\text{ is the no}\text{. of digital audio players}\\ \text{Monthly profit is at least }=\$100000\\ P\left(x\right)\ge 100000\\ P\left(x\right)=x\left(-27.5x+3520\right)+20000\ge 100000\\ \Rightarrow x\left(-27.5x+3520\right)+20000-100000\ge 0\\ \Rightarrow -27.5x^2+3520x-80000\ge 0\\ \Rightarrow x=\frac{-3250\pm \sqrt{{3520}^2-4\left(-27.5\right)\left(-80000\right)}}{2\left(-27.5\right)}\\ \Rightarrow x=\frac{-3250\pm \sqrt{12390400-8800000}}{-55}\\ \Rightarrow x=\frac{3250+\sqrt{3590400}}{55}or\frac{3250-\sqrt{3590400}}{55}\\ \Rightarrow x=29.55or98.45\\ \text{For numbers of audio players}\\ x=29.55\times 1000=29550\\ or\\ =98.45\times 1000=98450\end{array}$

Conclusion:

Thus the number of audio player is $29550or98450$

(c)

To determine

To Find: The total number of audio player to be sold to get the profit.

Answer to Problem 55PPS

  $47330or80670$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Function used by manufacturer to model monthly profit is }P\left(x\right)=x\left(-27.5x+3520\right)+20000\\ \text{Where }x\text{ is the no}\text{. of digital audio players}\\ \text{Monthly profit is at least }=\$100000\\ \text{Additional monthly expense of seller}=\$25000\end{array}$

Formula Used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $\begin{array}{l}\text{Function used by manufacturer to model monthly profit is }P\left(x\right)=x\left(-27.5x+3520\right)+20000\\ \text{Where }x\text{ is the no}\text{. of digital audio players}\\ \text{Monthly profit is at least }=\$100000\\ \text{Additional monthly expense of seller}=\$25000\\ P\left(x\right)\ge 100000\\ P\left(x\right)=x\left(-27.5x+3520\right)+20000\ge 100000\\ \Rightarrow x\left(-27.5x+3520\right)+20000-100000\ge 0\\ \Rightarrow -27.5x^2+3520x-80000\ge 0\\ \text{Because of additional expense of }\$25000\\ \Rightarrow -27.5x^2+3520x-105000\ge 0\\ \Rightarrow x=\frac{-3250\pm \sqrt{{3520}^2-4\left(-27.5\right)\left(-105000\right)}}{2\left(-27.5\right)}\\ \Rightarrow x=\frac{-3250\pm \sqrt{12390400-11550000}}{-55}\\ \Rightarrow x=\frac{3250+\sqrt{840400}}{55}or\frac{3250-\sqrt{8400400}}{55}\\ \Rightarrow x=47.33or80.67\\ \text{For numbers of audio players}\\ x=47.33\times 1000=47330\\ or\\ =80.67\times 1000=80670\end{array}$

  

Conclusion:

Thus the number audio player to be sold is $47330or80670$ .