Question

A rectangular field is enclosed on three sides using 1200m of fencing. The remaining side is formed by an existing wall (no fence needed). What dimensions would enclose 180, 000m² of land?

Answer 1

The dimensions that would enclose 180,000 m² of land are \( l = 300 \) meters and
`\( w = 600 \)` meters.

Let the length of the rectangle be \( l \) and the width be \( w \). The perimeter is given by \( P = 2l + w = 1200 \) since there are three sides enclosed with fencing.

The area \( A \) is given by \( A = lw \). We are given that \( lw = 180,000 \) and
`\( 2l + w = 1200 \).` We can solve these two equations simultaneously.

From the perimeter equation, we can express \( l \) in terms of \( w \):
\[ l = \frac{1200 - w}{2} \]

Now substitute this expression for \( l \) into the area equation:

`\[ (1200 - w)/(2) \cdot w = 180,000 \]`

Multiply both sides by 2 to simplify:
\[ (1200 - w) \cdot w = 360,000 \]

Expand and rearrange the equation:

`\[ 1200w - w^2 = 360,000 \]`

Bring all terms to one side to form a quadratic equation:
\[ w^2 - 1200w + 360,000 = 0 \]

Now, solve for \( w \) using the quadratic formula:

`\[ w = (-b \pm √(b^2 - 4ac))/(2a) \]For this equation, \( a = 1 \), \( b = -1200 \), and \( c = 360,000 \).\[ w = (1200 \pm √((-1200)^2 - 4(1)(360,000)))/(2(1)) \]\[ w = (1200 \pm √(1,440,000 - 1,440,000))/(2) \]\[ w = (1200 \pm √(0))/(2) \]\[ w = (1200)/(2) \]\[ w = 600 \]`

Now that we have \( w \), substitute it back into the expression for \( l \):

`\[ l = (1200 - 600)/(2) = 300 \]`