The velocity $v$ (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time $t$ (in minutes) as follows: The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson's $1/3^{\text{rd}}$ rule is _________.

The distance covered is given by the integral of velocity with respect to time:

The motorbike starts from rest, so at $t_0 = 0$, $v_0 = 0$. The data points with equal step size $h = 2$ minutes are:

• $t_0 = 0,\; v_0 = 0$

• $t_1 = 2,\; v_1 = 10$

• $t_2 = 4,\; v_2 = 18$

• $t_3 = 6,\; v_3 = 25$

• $t_4 = 8,\; v_4 = 29$

• $t_5 = 10,\; v_5 = 32$

• $t_6 = 12,\; v_6 = 20$

• $t_7 = 14,\; v_7 = 11$

• $t_8 = 16,\; v_8 = 5$

• $t_9 = 18,\; v_9 = 2$

• $t_{10} = 20,\; v_{10} = 0$

Using Simpson's $1/3^{\text{rd}}$ rule with $n = 10$ (an even number of intervals):

Sum of odd-indexed terms:

Sum of even-indexed terms (excluding boundaries):

Substituting these values into the formula:

Rounding to two decimal places, the distance is 309.33 km (official acceptable range: 308 to 310).