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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.
Solution :
Given:
Perimeter of the rectangular park ($P$) = $80$ m.
Area of the rectangular park ($A$) = $400$ m$^2$.
To Find:
Determine if such a park can be designed, and if so, find its length ($l$) and breadth ($b$).
Step 1: Formulating the Equations
Let the length of the rectangular park be $l$ meters and the breadth be $b$ meters.
The formula for the perimeter of a rectangle is $P = 2(l + b)$.
Given $P = 80$, we have:
$2(l + b) = 80$
$l + b = 40$
$b = 40 - l$ --- (Equation 1)
The formula for the area of a rectangle is $A = l \times b$.
Given $A = 400$, we have:
$l \times b = 400$ --- (Equation 2)
Step 2: Substituting Equation 1 into Equation 2
Substitute $b = 40 - l$ into the area equation:
$l(40 - l) = 400$
$40l - l^2 = 400$
Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:
$l^2 - 40l + 400 = 0$
Step 3: Checking the Discriminant
To determine if the roots are real, we calculate the discriminant ($D = b^2 - 4ac$).
Here, $a = 1$, $b = -40$, and $c = 400$.
$D = (-40)^2 - 4(1)(400)$
$D = 1600 - 1600$
$D = 0$
[Since $D = 0$, the quadratic equation has two equal real roots, meaning it is possible to design such a park.]
Step 4: Solving for $l$
Using the quadratic formula $l = \frac{-b \pm \sqrt{D}}{2a}$:
$l = \frac{-(-40) \pm \sqrt{0}}{2(1)}$
$l = \frac{40}{2}$
$l = 20$ m
Step 5: Finding the breadth $b$
Substitute $l = 20$ into Equation 1:
$b = 40 - 20$
$b = 20$ m
Conclusion:
Since the length and breadth are equal, the rectangular park is a square with side length $20$ m.
Final Answer: Yes, it is possible to design the park. The length is 20 m and the breadth is 20 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) $kx (x – 2) + 6 = 0$
- 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.
- 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 $48$.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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