Question

A rectangular garden is 15 m long and 8 m wide. It is surrounded by a path of uniform width. Given that the area of the paths is 50 m 2 : Find the width of the path.​

Answer 1

Given,

Length of garden $\[ = 15m\]$

Width of garden $\[ = 8m\]$

Area of the path $\[ = 50{m^2}\]$

Solution,

Consider the width of the path is b.

Observe the figure 2. Total area of the path,

= area of (1+2+3+4) section + area of ABCD + area of EFGH + area of CEIJ + area of DFKL

$\[\begin{array}{l}{b^2} + {b^2} + {b^2} + {b^2} + 15b + 15b + 8b + 8b = 50\\ \Rightarrow 4{b^2} + 46b - 50 = 0\\ \Rightarrow 2{b^2} + 23b - 25 = 0\\ \Rightarrow 2{b^2} + 25b - 2b - 25 = 0\end{array}\]$

Solve the quadratic equation,

$\[\begin{array}{l} \Rightarrow 2b\left( {b - 1} \right) + 25\left( {b - 1} \right) = 0\\ \Rightarrow \left( {2b + 25} \right)\left( {b - 1} \right) = 0\\ \Rightarrow b = 1, - \frac{{25}}{2}\end{array}\]$

Know that distance can not be negative so width will be 1m

Hence the width of path is 1m.