a rectangular field has an area of 1,764 m². the width of the field is 13 m more than the length. what is…

Explanation:

Step1: Let the length be $x$ m.

The width is $(x + 13)$ m. The area formula for a rectangle is $A=\text{length}\times\text{width}$. So we have the equation $x(x + 13)=1764$.

Step2: Expand the equation.

$x^{2}+13x-1764 = 0$.

Step3: Solve the quadratic - equation.

For a quadratic equation $ax^{2}+bx + c=0$ ($a = 1$, $b = 13$, $c=-1764$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(13)^{2}-4\times1\times(-1764)=169 + 7056=7225$. Then $x=\frac{-13\pm\sqrt{7225}}{2}=\frac{-13\pm85}{2}$. We get two solutions: $x_1=\frac{-13 + 85}{2}=\frac{72}{2}=36$ and $x_2=\frac{-13 - 85}{2}=\frac{-98}{2}=-49$. Since the length cannot be negative, $x = 36$ m.

Step4: Find the width.

The width is $x + 13=36+13 = 49$ m.

Step5: Calculate the perimeter.

The perimeter formula for a rectangle is $P = 2(\text{length}+\text{width})$. So $P=2(36 + 49)=2\times85 = 170$ m.

Answer:

170 m