A high-speed train X takes 32 seconds to completely pass a signal post, and it requires 72 seconds to travel through a tunnel that is 1200 meters long. Another train Y, which has a length of 1280 meters, travels at a constant speed of 72 km/hr. If both trains are moving along parallel tracks in the same direction, calculate the total duration in seconds needed for train X to completely overtake train Y.

The correct option is 224 seconds.

Step 1: Determine the length and speed of train X.

Let $L_X$ be the length of train X in meters, and $v_X$ be its speed in m/s.

When train X passes a stationary signal post, it covers a distance equal to its own length in 32 seconds:

$L_X=32\times v_X$

When train X travels through a tunnel of length 1200 meters, the total distance covered is the sum of the train's length and the tunnel's length ($L_X+1200$), taking 72 seconds:

$L_X+1200=72\times v_X$

Substitute $L_X=32v_X$ into the equation above:

$32v_X+1200=72v_X$

$72v_X-32v_X=1200$

$40v_X=1200$

$v_X=\frac{1200}{40}=30\text{m/s}$

Now, calculate the length of train X:

$L_X=32\times 30=960\text{meters}$

Step 2: Determine the length and speed of train Y.

Length of train Y, $L_Y=1280\text{meters}$.

Speed of train Y, $v_Y=72\text{km/hr}$.

Convert the speed of train Y to m/s:

$v_Y=72\times \frac{5}{18}=20\text{m/s}$

Step 3: Calculate the time taken for train X to completely overtake train Y.

When train X overtakes train Y while moving in the same direction, the relative speed is the difference between their speeds:

$v_{\text{relative}}=v_X-v_Y=30-20=10\text{m/s}$

The total distance to be covered for a complete overtake is the sum of the lengths of both trains:

$D_{\text{total}}=L_X+L_Y=960+1280=2240\text{meters}$

Now, calculate the time required:

$\text{Time}=\frac{D_{\text{total}}}{v_{\text{relative}}}=\frac{2240}{10}=224\text{seconds}$

Thus, the total duration needed for train X to completely overtake train Y is 224 seconds.