**Topic**

One wants to design a rectangular flower bed inscribed in a circular plot of land with a diameter of 50 m (Figure 23). Determine the size of the rectangular flower garden so that the total distance traveled around the flower garden is 140 m.

**Solution method – See details**

Set the length of one side of the rectangle to \(x\)(m) (\(0 < x < 50\)).

Express the remaining side and perimeter of the rectangle in terms of x.

**Detailed explanation**

Set the length of one side of the rectangle to \(x\)(m) (\(0 < x < 50\)).

The length of the diagonal of the rectangle = Diameter of the circle = 50m.

The length of the remaining side of that rectangle is \(\sqrt {{{50}^2} – {x^2}} = \sqrt {2500 – {x^2}} \) (m)

Then, the total distance traveled around the flower garden equal to the perimeter of the rectangle is: \(2\left( {\sqrt {2500 – {x^2}} + x} \right) = 140\) (m)

We have the equation: \(2\left( {\sqrt {2500 – {x^2}} + x} \right) = 140 \Leftrightarrow \sqrt {2500 – {x^2}} + x = 70 \Rightarrow \sqrt {2500 – {x^2}} = 70 – x\)

\(\begin{array}{l} \Leftrightarrow \left\{ \begin{array}{l}x > 0\\70 – x \ge 0\\2500 – {x^2} = {\left( { 70 – x} \right)^2}\end{array} \right \Leftrightarrow \left\{ \begin{array}{l}0 < x \le 70\\2500 – {x^2} = {x ^2} – 140x + {70^2}\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}0 < x \le 70\\2{x^2} – 140x + 2400 = 0\end{array} \right \Leftrightarrow \left\{ \begin{array}{l}0 < x \le 70\\\left[\begin{array}{l}x=30\\x=40\;\end{array}\right\quad\end{array}\right\end{array}\)[\begin{array}{l}x=30\x=40\;\end{array}\right\quad\end{array}\right\end{array}\)

If \(x = 40\) then the remaining side length is 30 (m) and vice versa.

So the size of the flower garden is 30 x 40 (m)