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Q3:
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800$ $m^2$? If so, find its length and breadth.
Solution :
Given:
1. The shape of the mango grove is rectangular.
2. The length ($l$) of the grove is twice its breadth ($b$), i.e., $l = 2b$.
3. The area ($A$) of the rectangular grove is $800$ $m^2$.
To Find:
1. Determine if such a rectangular grove can be designed.
2. If possible, find the length ($l$) and breadth ($b$).
Visual Representation:
Step 1: Formulating the Algebraic Equation
Let the breadth of the rectangular mango grove be $b$ meters.
According to the problem, the length $l$ is twice the breadth, so $l = 2b$ meters.
The formula for the area of a rectangle is: $Area = length \times breadth$.
Substituting the given values into the formula:
$800 = (2b) \times (b)$
$800 = 2b^2$
Step 2: Simplifying the Quadratic Equation
Divide both sides of the equation by $2$:
$\frac{800}{2} = \frac{2b^2}{2}$
$400 = b^2$
Rearranging into the standard quadratic form $ax^2 + bx + c = 0$:
$b^2 - 400 = 0$
Step 3: Determining Possibility using the Discriminant
To check if real roots exist, we calculate the discriminant ($D$) where $D = B^2 - 4AC$.
In the equation $b^2 + 0b - 400 = 0$, we have $A = 1$, $B = 0$, and $C = -400$.
$D = (0)^2 - 4(1)(-400)$
$D = 0 + 1600$
$D = 1600$
[Since $D > 0$, the quadratic equation has two distinct real roots, meaning it is possible to design the grove.]
Step 4: Solving for Breadth and Length
From $b^2 = 400$, we take the square root of both sides:
$b = \pm \sqrt{400}$
$b = \pm 20$
[Since breadth cannot be negative, we discard $b = -20$.]
Therefore, $b = 20$ meters.
Now, calculate the length $l$ using $l = 2b$:
$l = 2 \times 20 = 40$ meters.
Final Answer:
Yes, it is possible to design the mango grove. The breadth of the grove is $20$ m and the length is $40$ m.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.3
- Q1(i): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2x^2 – 3x + 5 = 0$
- Q1(ii): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (ii) $3x^2 – 4\sqrt{3}x + 4 = 0$
- Q1(iii): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (iii) $2x^2 – 6x + 3 = 0$
- Q2(i): Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (i) $2x^2 + kx + 3 = 0$
- Q2(ii): Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (ii) $kx (x – 2) + 6 = 0$
- Q4: Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is $20$ years. Four years ago, the product of their ages in years was $48$.
- Q5: Is it possible to design a rectangular park of perimeter $80$ m and area $400$ $m^2$? If so, find its length and breadth.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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