Problem Statement: A farmer wishes to start 300 square metre rectangular vegetable Garden. since he has only 30 m Barbed wire he fence the sides of rectangular Garden letting his house compound well act as the fourth fence. Find the dimensions of his garden? Explain in details.

Solution:

To find the dimensions of the rectangular garden, we need to use the given information about the area of the garden and the length of the barbed wire available to fence the garden.

Step 1: Find the Perimeter of the Rectangular Garden

Since the length of the barbed wire available is 30 m, we can use this information to find the perimeter of the rectangular garden. The perimeter can be calculated as:

Perimeter = 2 (length + width)

We know that one side of the garden is already fenced by the farmer's house compound wall, so we only need to fence the other three sides. Therefore, the perimeter of the garden is:

30 = 2 (length + width) + length

Simplifying the equation, we get:

30 = 3 length + 2 width

Step 2: Find the Area of the Rectangular Garden

We know that the area of the rectangular garden is 300 square metres. Therefore, we can write:

Area = length × width = 300

Step 3: Solve the Equations Simultaneously

We have two equations in two variables (length and width) that we can solve simultaneously to find the dimensions of the garden. Substituting the value of area from the second equation into the first equation, we get:

30 = 3 length + 2 width

300 = length × width

Now, we can use substitution or elimination to solve for one variable and then find the other variable. Using substitution, we can isolate one variable in terms of the other and substitute into the other equation.

From the first equation, we can write:

length = (30 - 2 width) / 3

Substituting this value into the second equation, we get:

300 = (30 - 2 width) / 3 × width

Simplifying the equation, we get:

100 = (10 - 2/3 width) × width

Multiplying out the brackets, we get:

100 = 10 width - 2/3 width²

Multiplying both sides by 3, we get:

300 = 30 width - 2 width²

Rearranging the equation, we get:

2 width² - 30 width + 300 = 0

Dividing both sides by 2, we get:

width² - 15 width + 150 = 0

Using the quadratic formula, we get:

width = (15 ± √(15² - 4×1×150)) / 2

width = (15 ± √225) / 2

width = 7.5 or 20

Since the width cannot be larger than the length, we can ignore the solution of width = 20. Therefore, the width of the garden is:

width = 7.5 m

Using the first equation, we can find the length:

30 = 3 length + 2 × 7.5

30 - 15 = 3 length

length = 5 m

Therefore, the dimensions of