Question 3: Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 square meter? If so, find its length and breadth.

**Solution:** Let us assume, breadth = x, then length = 2x

As per question;

`2x^2 = 800`

Or, `x^2 = 400`

Or, `x = 20`

Hence, length = 40 m and breadth = 20 m

Question 4: 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.

**Solution:** Let us assume, age of one friend = x, then age of another friend `= 20 – x`

Four years ago, age of first friend `= x – 4`

Four years ago, age of second friend `= 16 – x`

As per question;

`(x – 4)(16 – x) = 48`

Or, `16x – x^2 – 64 + 4x = 48`

Or, `20x – x^2 – 64 – 48 = 0`

Or, `20x – x^2 – 112 = 0`

Or, `x^2 – 20x + 112 = 0`

Let us check the existence of root;

`D=b^2-4ac`

`=(-20)^2-4xx112`

`=400xx448=-48`

Here; D < 0, hence no real root is possible. The given situation is not possible.

Question 5: Is it possible to design a rectangular park of perimeter 80m and area 400 square meter? If so, find its length and breadth.

**Solution:** `text(Perimeter) = 2text((length + breadth))`

Or, `2text((length + breadth)) = 80 m`

Or, `text(length + breadth) = 40 m`

If length is assumed to be x, then breadth `= 40 – x`

As per question;

`x(40 – x) = 400`

Or, `40x – x^2 = 400`

Or, `40x – x^2 – 400 = 0`

Or, `x^2 – 40x + 400 = 0`

Let us check the existence of roots:

`D=b^2-4ac`

`=(-40)^2-4xx100`

`=1600-1600=0`

Here; D = 0, hence roots are possible.

Now, root can be calculated as follows:

`text(Root)=(-b)/(2a)=(40)/(2)=20 m`

This is a square with side 20 m

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