Chapter 4.8, Problem 45PPS

(a)

To determine

To Draw: The graph of the quadratic inequality.

Explanation of Solution

Given:

  

  $\begin{array}{l}\text{Length of the swimming pool=}24ft\\ \text{Breadth of the swimming pool}=12ft\\ \text{Width of the deck=}x\\ \text{Maximum area of the pool and deck}=750f{t}^2\end{array}$

Formula Used:

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ areaof\text{rectangle=}l\times b\end{array}$

Calculation:

  $\begin{array}{l}\text{Length of the swimming pool=}24ft\\ \text{Breadth of the swimming pool}=12ft\\ \text{Width of the deck=}x\\ \text{Maximum area of the pool and deck}=750f{t}^2\\ \text{New length of pool including deck}=\left(24+x\right)ft\\ \text{New breadth of the pool including deck}=\left(12+x\right)ft\\ \text{New area of the pool including deck=}\left(24+x\right)ft\times \left(12+x\right)ft=\left(x^2+36x+288\right)f{t}^2\\ But\\ x^2+36x+288<750\\ \Rightarrow x^2+36x+288-750<0\\ \Rightarrow x^2+36x-462<0\\ \Rightarrow x=\frac{-36\pm \sqrt{{\left(36\right)}^2-4\left(1\right)\left(-462\right)}}{2\left(1\right)}\\ \Rightarrow x=\frac{-36\pm \sqrt{1296+1848}}{2}\\ \Rightarrow x=\frac{-36k\pm 2\sqrt{786}}{2}\\ \Rightarrow x=-18-\sqrt{786}or-18+\sqrt{786}\end{array}$

  

Conclusion:

(b)

To determine

Answer to Problem 45PPS

  $-18-\sqrt{786}or-18+\sqrt{786}$

Explanation of Solution

Given:

  

  $\begin{array}{l}\text{Length of the swimming pool=}24ft\\ \text{Breadth of the swimming pool}=12ft\\ \text{Width of the deck=}x\\ \text{Maximum area of the pool and deck}=750f{t}^2\end{array}$

Formula Used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $\begin{array}{l}\text{Length of the swimming pool=}24ft\\ \text{Breadth of the swimming pool}=12ft\\ \text{Width of the deck=}x\\ \text{Maximum area of the pool and deck}=750f{t}^2\\ \text{New length of pool including deck}=\left(24+x\right)ft\\ \text{New breadth of the pool including deck}=\left(12+x\right)ft\\ \text{New area of the pool including deck=}\left(24+x\right)ft\times \left(12+x\right)ft=\left(x^2+36x+288\right)f{t}^2\\ But\\ x^2+36x+288<750\\ \Rightarrow x^2+36x+288-750<0\\ \Rightarrow x^2+36x-462<0\\ \Rightarrow x=\frac{-36\pm \sqrt{{\left(36\right)}^2-4\left(1\right)\left(-462\right)}}{2\left(1\right)}\\ \Rightarrow x=\frac{-36\pm \sqrt{1296+1848}}{2}\\ \Rightarrow x=\frac{-36k\pm 2\sqrt{786}}{2}\\ \Rightarrow x=-18-\sqrt{786}or-18+\sqrt{786}\end{array}$

Conclusion:

Thus the new dimensions would be $-18-\sqrt{786}or-18+\sqrt{786}$