Question

a rectangular field has an area of 1,764 m². the width of the field is 13 m more than the length. what is the perimeter of the field?
121 m
144 m
168 m
170 m

Explanation:

Step1: Define variables

Let the length of the rectangular field be \( l \) meters. Then the width \( w = l + 13 \) meters.

Step2: Use area formula

The area of a rectangle is \( A = l\times w \). Given \( A = 1764 \, m^2 \), so \( l(l + 13)=1764 \).
Expanding, we get \( l^{2}+13l - 1764 = 0 \).

Step3: Solve quadratic equation

Using the quadratic formula \( l=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for \( ax^{2}+bx + c = 0 \). Here, \( a = 1 \), \( b = 13 \), \( c=- 1764 \).
First, calculate the discriminant \( D=b^{2}-4ac=(13)^{2}-4\times1\times(-1764)=169 + 7056 = 7225 \).
Then \( \sqrt{D}=\sqrt{7225}=85 \).
So \( l=\frac{-13\pm85}{2} \). We take the positive root: \( l=\frac{-13 + 85}{2}=\frac{72}{2}=36 \) meters.

Step4: Find width

Width \( w = l + 13=36 + 13 = 49 \) meters.

Step5: Calculate perimeter

Perimeter of a rectangle \( P = 2(l + w) \). Substituting \( l = 36 \) and \( w = 49 \), we get \( P = 2(36 + 49)=2\times85 = 170 \) meters.

Answer:

170 m