A rectangular garden is 42 m long and 28 m wide. A path of uniform width x m is constructed outside it on all four sides. The total area of the garden and the path is 1848 square metres. What is the value of x?
Answer and explanation
Correct answer: \(\frac{-35+\sqrt{1897}}{2}\) m
The outer dimensions, including the path, are \((42+2x)\) m and \((28+2x)\) m. Thus, \((42+2x)(28+2x)=1848\), which gives \(4x^2+140x-672=0\), or \(x^2+35x-168=0\). Hence, \(x=\frac{-35\pm\sqrt{1897}}{2}\). Since a width must be positive, the valid value is \(x=\frac{-35+\sqrt{1897}}{2}\approx4.28\) m. The negative root is not physically meaningful. Exam tip: when a path surrounds a rectangle externally, add \(2x\) to both its length and breadth.
Frequently asked questions
What is the correct answer to this question?
\(\frac{-35+\sqrt{1897}}{2}\) m
Why is this the correct answer?
The outer dimensions, including the path, are \((42+2x)\) m and \((28+2x)\) m. Thus, \((42+2x)(28+2x)=1848\), which gives \(4x^2+140x-672=0\), or \(x^2+35x-168=0\). Hence, \(x=\frac{-35\pm\sqrt{1897}}{2}\). Since a width must be positive, the valid value is \(x=\frac{-35+\sqrt{1897}}{2}\approx4.28\) m. The negative root is not physically meaningful. Exam tip: when a path surrounds a rectangle externally, add \(2x\) to both its length and breadth.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Word Problems and Applications.
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