Question

a rectangular playing field is 18km wide. it is surrounded by a path 6m wide such that its area equal to the area of the path. calculate the length field.

Explanation:

Step1: Define variables

Let the length of the rectangular playing field be \( l \) km. The width of the field is \( 18 \) km. The path is \( 6 \) m wide, but we need to convert units to be consistent. Wait, maybe it's a typo and should be \( 6 \) km? Assuming it's \( 6 \) km (since \( 6 \) m would be too small relative to \( 18 \) km). The outer rectangle (field + path) has length \( l + 2\times6=l + 12 \) km and width \( 18+2\times6 = 30 \) km.

Step2: Calculate areas

Area of the field: \( A_{field}=l\times18 = 18l \) square km.
Area of the outer rectangle: \( A_{outer}=(l + 12)\times30=30l+360 \) square km.
Area of the path: \( A_{path}=A_{outer}-A_{field}=30l + 360-18l=12l + 360 \).

Step3: Set up equation

Given that area of the field equals area of the path:
\( 18l=12l + 360 \)

Step4: Solve for \( l \)

Subtract \( 12l \) from both sides:
\( 18l-12l=360 \)
\( 6l = 360 \)
Divide both sides by \( 6 \):
\( l=\frac{360}{6}=60 \)

Answer:

The length of the field is \( 60 \) km.