Problem Analysis:

The problem involves calculating the increase in the height of water in a tank after a certain volume of water is pumped into it. The tank is in the shape of a rectangular prism, so the relationship between volume, base area, and height will be used.

Given Data:

1. Base dimensions of the tank:
   • Length = $6 \, \text{meters}$,
   • Width = $4 \, \text{meters}$,
   • Base area = $6 \times 4 = 24 \, \text{square meters}$.
2. Water usage in the morning: $120 \, \text{cubic meters}$.
3. Additional water pumped into the tank in the afternoon:
   • $125\%$ more than the morning usage,
   • Volume pumped = $120 + 1.25 \times 120 = 120 + 150 = 270 \, \text{cubic meters}$.

Goal:

Find the increase in the height of water after $270 \, \text{cubic meters}$ is pumped into the tank.

Solution:

We use the formula for the volume of a rectangular prism: $\text{Volume} = \text{Base Area} \times \text{Height}.$ Rearranging for the height: $\text{Height} = \frac{\text{Volume}}{\text{Base Area}}.$

1. Calculate the height increase:

   • Volume pumped = $270 \, \text{cubic meters}$,
   • Base area = $24 \, \text{square meters}$,
   • Increase in height: $\Delta h = \frac{270}{24} = 11.25 \, \text{meters}.$

2. Express $11.25 \, \text{meters}$ in mixed fractions:

   $$$$

Given Data:

1. The base area of the tank: [ \text{Base Area} = 6 , \text{meters} \times 4 , \text{meters} = 24 , \text{square meters}.$$

2. Water used in the morning = $120 \, \text{cubic meters}$.

3. In the afternoon, $125\%$ more than the morning's water was pumped into the tank: $\text{Afternoon water} = 120 + 1.25 \times 120 = 120 + 150 = 270 \, \text{cubic meters}.$

4. Total water volume added = $270 \, \text{cubic meters}$.

Goal:

Find the increase in the height of the water after $270 \, \text{cubic meters}$ was pumped into the tank.

Formula:

The volume of a rectangular prism is: $\text{Volume} = \text{Base Area} \times \text{Height}.$ Rearranging for height: $\text{Height} = \frac{\text{Volume}}{\text{Base Area}}.$

Step-by-Step Calculation:

1. Substitute the known values: $\Delta h = \frac{270 \, \text{cubic meters}}{24 \, \text{square meters}}.$

2. Perform the division: $\Delta h = 11.25 \, \text{meters}.$

Final Conversion to Mixed Fractions:

11.25 meters can be written as: $11.25 = 11 + \frac{1}{4} \, \text{meters}.$

Thus, the increase in height is: $\boxed{11 \frac{1}{4} \, \text{meters}}.$

Correct Answer: $D) 11 \frac{1}{4}$.

Any questions or would you like additional clarifications? Here are related questions for further practice:

1. What is the formula for the volume of a rectangular prism, and how does it relate to height and base area?
2. If 300 cubic meters of water were pumped into the tank instead, what would the height increase be?
3. How would the answer change if the tank had a different base width, e.g., $5 \, \text{meters}$?
4. Why does increasing the base area reduce the increase in height for the same volume of water?
5. Can you convert $11.25$ meters to decimal feet?

Tip:

When working with percent increases, always calculate the increase first, then add it to the original value.