Step 1 :The total area of the rectangular plot is given as 1050 m². This can be represented as the product of the length and the width, or \(x*y = 1050\).
Step 2 :The total length of the fencing is the perimeter of the rectangle plus the length of the fence dividing the plot into two, which is \(2x + 3y\).
Step 3 :We want to minimize the amount of fencing, so we need to minimize the function \(2x + 3y\).
Step 4 :We can express y in terms of x using the area equation: \(y = 1050/x\).
Step 5 :Substitute \(y = 1050/x\) into the fencing function: \(2x + 3(1050/x) = 2x + 3150/x\).
Step 6 :To find the minimum of this function, we can take the derivative and set it equal to zero.
Step 7 :The derivative of \(2x + 3150/x\) is \(2 - 3150/x²\).
Step 8 :Setting this equal to zero gives: \(2 - 3150/x² = 0\).
Step 9 :Solving for x gives: \(x² = 3150/2 = 1575\).
Step 10 :Taking the square root of both sides gives: \(x = \sqrt{1575} \approx 39.68 m\).
Step 11 :Substitute \(x = 39.68\) into \(y = 1050/x\) to find y: \(y = 1050/39.68 \approx 26.45 m\).
Step 12 :So, the dimensions that require the least amount of fencing are approximately 39.68 m by 26.45 m. This can be confirmed by checking that these dimensions give an area of 1050 m²: \(39.68 m * 26.45 m = 1050 m²\).
Step 13 :The final answer is \(\boxed{39.68 m, 26.45 m}\).