Quadratic Equations | Exercise 4.3

Question 3

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2800 \text{ m}^2? If so, find its length and breadth.

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Solution
Understand the Question
  • Let the breadth of the rectangular mango grove be x mx\text{ m}.
  • Since the length is twice the breadth, the length is 2x m2x\text{ m}.
  • The area of a rectangle is Length×Breadth=2x×x=2x2\text{Length} \times \text{Breadth} = 2x \times x = 2x^2.
  • Equating this to the given area of 800 m2800\text{ m}^2, we check if a valid real, positive value for xx exists. If x>0x > 0, designing the grove is possible.

Step 1 · Set Up the Equation

Let the breadth of the rectangular grove be x mx\text{ m}.

Then, the length =2x m= 2x\text{ m}.Diagram 1

Area=Length×Breadth=2x×x=2x2\text{Area} = \text{Length} \times \text{Breadth} = 2x \times x = 2x^2

Given that the area is 800 m2800\text{ m}^2 2x2=8002x^2 = 800

Step 2 · Solve for Breadth (xx)

x2=8002=400\begin{aligned} x^2 &= \dfrac{800}{2} \\[0.6em] &= 400 \end{aligned}

Taking the square root on both sides

x=±400=±20\begin{aligned} x &= \pm \sqrt{400} \\[0.6em] &= \pm 20 \end{aligned}

Since breadth represents a physical distance, it cannot be negative. Breadth=20 m\text{Breadth} = 20\text{ m}

Step 3 · Find the Length

Length=2x=2×20=40 m\begin{aligned} \text{Length} &= 2x \\[0.6em] &= 2 \times 20 \\[0.6em] &= 40\text{ m} \end{aligned}

Since real positive values exist for both length and breadth, it is possible to design such a grove.

Answer

Yes, it is possible to design the mango grove.

  • Length=40 m\text{Length} = 40\text{ m}
  • Breadth=20 m\text{Breadth} = 20\text{ m}
Common Mistakes
  • Negative Root Error: Considering both ±20\pm 20 as dimensions. Physical measurements like length and breadth can only be positive (x>0x > 0).
  • Confusing Area with Perimeter: Writing 2(x+2x)=8002(x + 2x) = 800 instead of x×2x=800x \times 2x = 800.
  • Swapping Dimensions: Stating length as 20 m20\text{ m} and breadth as 40 m40\text{ m}, when the length is twice the breadth.

More questions in Exercise 4.3

Q1

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x2−3x+5=02x^2 - 3x + 5 = 0

(ii) 3x2−43x+4=03x^2 - 4\sqrt{3}x + 4 = 0

(iii) 2x2−6x+3=02x^2 - 6x + 3 = 0

Q2

Find the values of kk for each of the following quadratic equations, so that they have two equal roots.

(i) 2x2+kx+3=02x^2 + kx + 3 = 0

(ii) kx(x−2)+6=0kx (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 m2800 \text{ 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 2020 years. Four years ago, the product of their ages in years was 4848.

Q5

Is it possible to design a rectangular park of perimeter 80 m80\text{ m} and area 400 m2400\text{ m}^2? If so, find its length and breadth.

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