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.

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.