The sides of a cyclic quadrilateral are in A.P., the shortest is 6 and the difference of longest and the shortest is also 6. If P is the square of the area of the quadrilateral then P/1440 is ___

Let's solve the problem step by step:

1. Let's assume the sides of the cyclic quadrilateral are a, a+d, a+2d, and a+3d, where a is the shortest side and d is the common difference in the arithmetic progression.

2. Given that the shortest side is 6, we have a = 6.

3. The difference between the longest side and the shortest side is also 6. So, (a+3d) - a = 6.

   Simplifying this equation, we get: 3d = 6 d = 2

4. Now, we can find the lengths of the sides of the quadrilateral: a = 6 a+d = 6+2 = 8 a+2d = 6+4 = 10 a+3d = 6+6 = 12

5. To find the area of the cyclic quadrilateral, we can use Brahmagupta's formula: Area = √((s-a)(s-b)(s-c)(s-d)) where s is the semiperimeter and a, b, c, d are the lengths of the sides.

   The semiperimeter, s, is given by: s = (a + b + c + d)/2

   Substituting the values, we get: s = (6 + 8 + 10 + 12)/2 = 36/2 = 18

   Now, we can calculate the area: Area = √((18-6)(18-8)(18-10)(18-12)) = √(12 * 10 * 8 * 6) = √(5760) = 76.1577310586 (approx.)

6. Finally, we need to find P, which is the square of the area: P = (76.1577310586)^2 = 5799.99999999 (approx.)

7. To find P/1440, we divide P by 1440: P/1440 = 5799.99999999 / 1440 = 4.02777777777 (approx.)

Therefore, P/1440 is approximately 4.0278.