Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Step-by-Step Solution
Key Concept: Translate the geometric condition into an algebraic equation. Let the breadth be \(x\) metres; then the length is \(2x\) metres. Using the area formula \(\text{Area}=\text{length}\times\text{breadth}\) gives a quadratic equation in \(x\). Solve the quadratic and select the positive root because dimensions cannot be negative.
1. Let the breadth be \(x\) metres.
Then the length = \(2x\) metres (given).\
2. Write the area condition:
\[\text{Area}=\text{length}\times\text{breadth}=2x\times x = 2x^{2}=800\]\
3. Form the quadratic equation:
\[2x^{2}=800 \;\Rightarrow\; x^{2}=400\]
This can be written as \[x^{2}-400=0\], which is a quadratic in \(x\).\
4. Solve the quadratic:
\[x^{2}=400 \;\Rightarrow\; x=\pm\sqrt{400}=\pm20\]
Since a length cannot be negative, we take \(x=20\) metres.\
5. Find the length:
Length = \(2x = 2\times20 = 40\) metres.\
6. Conclusion:
The required rectangular mango grove can be designed with breadth \(20\) m and length \(40\) m.
Correct Answer: Length = 40 m, Breadth = 20 m (possible).