Step 1: Calculate the volume of water required in the tank

The tank has a rectangular base of dimensions $200\text{ m} \times 150\text{ m}$, and the water level needs to rise by $2\text{ m}$. \nVolume $V_{\text{tank}} = \text{Base Area} \times \text{Height} = 200 \times 150 \times 2 = 60,000\text{ m}^3$.

$V_{\text{tank}} = 200 \times 150 \times 2 = 60,000\text{ m}^3$

Step 2: Convert the speed of water from km/h to m/min

The speed of water flowing through the pipe is given as $20\text{ km/h}$. Converting this into meters per minute: \nSpeed $v = 20 \times \frac{1000\text{ m}}{60\text{ min}} = \frac{20000}{60} = \frac{1000}{3}\text{ m/min}$.

$v = 20 \times \frac{1000}{60} = \frac{1000}{3}\text{ m/min}$

Step 3: Calculate the volume of water flowing per minute

The cross-sectional area of the rectangular pipe is $1.5\text{ m} \times 1.25\text{ m}$. \nVolume of water flowing per minute = Area of cross-section $\times$ Speed per minute = $1.5 \times 1.25 \times \frac{1000}{3} = \frac{15}{10} \times \frac{125}{100} \times \frac{1000}{3} = 625\text{ m}^3\text{/min}$.

$\text{Volume per minute} = 1.5 \times 1.25 \times \frac{1000}{3} = 625\text{ m}^3\text{/min}$

Step 4: Find the total time required

Divide the total volume of water needed in the tank by the volume of water entering per minute: \n$\text{Time} = \frac{60000}{625} = 96\text{ minutes}$.

$\text{Time} = \frac{60000}{625} = 96\text{ minutes}$