Chapter 3.6, Problem 44E

To determine

To calculate: The possible widths of the fountain.

Answer to Problem 44E

The required width of the fountain is $\left[70,130\right]$ .

Explanation of Solution

Given information:

Perimeter of the rectangular fountain = 400 feet

Least area of the fountain = 9100 square feet

Formula used:

Perimeter of a rectangle:

Perimeter of a rectangle is $2\left(\text{length}\times \text{width}\right)$

Area of the rectangle:

Area of a rectangle is $\text{length}\times \text{width}$

Calculation:

Consider the width and length of the rectangular fountain be x and y .

The given area of the fountain is 900 feet.

So, $\begin{array}{c}\frac{2\left(x+y\right)}{2}=\frac{400}{2}\\ x+y=200\end{array}$

Rewrite the above equation as:

  $\begin{array}{c}x+y=200\\ y=200-x\end{array}$

The least area of the fountain is 9100 square feet.

So, $x\times y\ge 9100$

Substitute $200-x$ in the above inequality.

  $\begin{array}{c}x\left(200-x\right)\ge 200\\ 200x-x^2\ge 200\\ x^2-200x+9100\le 0\end{array}$

Solve the above inequality.

  $\begin{array}{c}{x}^2-200x+9100\le 0\\ x^2-130x-70x+9100\le 0\\ x\left(x-130x\right)-70\left(x-130\right)\le 0\\ \left(x-130\right)\left(x-70\right)\le 0\end{array}$

The solution of the inequality is $\left[70,130\right]$ .

Hence, the required width of the fountain is $\left[70,130\right]$ .