The area of a triangular field with sides 220 m, 231 m and 319 m is equal to 15 times the area of a circular garden . What is the diameter (in m) of the circular garden?

Explanation

Given: The area of a triangular field with sides 220 m, 231 m, and 319 m is equal to 15 times the area of a circular garden Formula used: s = (a + b + c)/2 Area of triangle = √[s (s – a) (s – b) (s – c)] where s → semi-perimeter of the triangle and a, b, c are the sides of the triangle. Calculation: s = (220 + 231 + 319)/2 ⇒ s = 770/2 ⇒ s = 385 Now, Area of triangle = √[385(385 - 220)(385 - 231)(385 - 319)] ⇒ √[385 × 165 × 154 × 66] ⇒ √[11 × 7 × 5 × 3 × 11 × 5 × 7 × 2 × 11 × 11 × 3 × 2] ⇒ 11 × 11 × 7 × 5 × 3 × 2 ⇒ 25410 m 2 According to the question, 25410 = 15 × πr 2 ⇒ 1694 = πr 2 ⇒ r 2 = 1694 × (7/22) ⇒ r 2 = 539 ⇒ r = √(49 × 11) ⇒ r = 7√11m So, the diameter of the garden = 2r =14√11 m ∴ The diameter (in m) of the circular garden is 14√11.