Question 5

Step 1 — Define dimensions

Let's define the length of the park as $L$.

Let's define the breadth of the park as $B$.

We know the perimeter is 80 m.

The perimeter formula is $2(L+B)$.

So, $2(L+B) = 80$.

This means $L+B = 40$.

Let the length be $\mathbf{x}$ meters.

Then the breadth will be $\mathbf{(40 - x)}$ meters.

$\boxed{\text{Breadth} = (40 - x) \text{ m}}$

Step 2 — Form the equation

We know the area is 400 m².

The area formula is $L \times B$.

So, $x(40 - x) = 400$.

Let's expand this equation.

$40x - x^2 = 400$

We will rearrange it into standard form.

$x^2 - 40x + 400 = 0$

$\boxed{x^2 - 40x + 400 = 0}$

Step 3 — Solve the equation

We need to solve this quadratic equation.

We can use the factorization method.

We look for two numbers.

They must multiply to 400.

They must add up to -40.

These numbers are -20 and -20.

$x^2 - 20x - 20x + 400 = 0$

Let's factor by grouping.

$x(x - 20) - 20(x - 20) = 0$

$(x - 20)(x - 20) = 0$

This can be written as a square.

$(x - 20)^2 = 0$

Let's find the value of $x$.

$x - 20 = 0$

$x = 20$

$\boxed{x = 20}$

Step 4 — Find length and breadth

We found the value of $x$.

This value is the length.

So, the length is 20 m.

Now, let's find the breadth.

Breadth is $\mathbf{(40 - x)}$.

Substitute $x = 20$ into the breadth.

$\text{Breadth} = 40 - 20$

$\text{Breadth} = 20 \text{ m}$

Since we found real values, it is possible.

$\boxed{\text{Length} = 20 \text{ m}, \text{Breadth} = 20 \text{ m}}$

Answer

(i) Yes, it is possible to design such a park. (ii) The length of the park is 20 m. (iii) The breadth of the park is 20 m.