Chapter 5, Problem 10STP

To determine

To calculate: The width of the walkway in feet:

The width of the stone path is $x$ feet wide around a rectangular flower garden. the width and the length of the garden is: $12\text{ feet and 25 feet}$ .

  

Answer to Problem 10STP

The width of the stone path is: $6.605$ .

Explanation of Solution

Given information:

The width of the walkway in feet:

The width of the stone path is $x$ feet wide around a rectangular flower garden. the width and the length of the garden is: $12\text{ feet and 25 feet}$ .

If the combined area of the garden and the stone path is $558$ square feet.

Calculation:

Consider the provided statement:

First find the area of the inner rectangle:

The length of the inner rectangle: $25\text{feet}$

The width of the inner rectangle: $\text{12feet}$

Then the area of the rectangle in general: $l\times b$

Then apply the values in the formula of area of rectangle:

  $\begin{array}{c}l\times b=25\times 12\\ =300\end{array}$

Now find the combined area of garden of the stone path: $588$ feet.

Now the new length of the combined rectangle is: $25+x$

Now the new width of the combined rectangle is: $12+x$

Apply the values in the formula of the area of rectangle:

  $\begin{array}{c}area=l\times b\\ 588=\left(25+x\right)\left(12+x\right)\\ 588=25\cdot 12+25x+12x+x^2\\ 588=x^2+37x+300\end{array}$

Simplified further:

  $\begin{array}{c}588=x^2+37x+300\\ 588-300=x^2+37x\\ x^2+37x=288\\ x^2+37x-288=0\end{array}$

Simplified the equation by : $x^2+37x-288=0$

By the further simplification use the quadratic method:

  $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2}\\ x=\frac{-37\pm \sqrt{{\left(37\right)}^2-4\cdot 1\cdot 288}}{2}\end{array}$

Take this term for positive:

  $\begin{array}{c}x=\frac{-37-\sqrt{2521}}{2}\\ x=-43.605\end{array}$

Take this term for negative:

  $\begin{array}{c}x=\frac{-37+\sqrt{2521}}{2}\\ x=6.605\end{array}$

So, width of the stone path is: $6.605$

Hence, the width of the stone path is: $6.605$ .