Let's work through the problem step-by-step:

Step 1: Calculate the Initial Water Level

The pond is shaped as a trapezoidal prism, with the following dimensions given:

• The top width is 2 meters.
• The bottom width is 1 meter.
• The height of the trapezoidal cross-section is 1.3 meters.
• The depth (perpendicular to the trapezoid) is 0.5 meters.

Step 2: Calculate the Area of the Trapezoidal Cross-Section

The area $A$ of a trapezoid with bases $a$ and $b$ and height $h$ is given by: $A = \frac{1}{2} \times (a + b) \times h$ Here:

• $a = 2$ meters
• $b = 1$ meter
• $h = 1.3$ meters

So, $A = \frac{1}{2} \times (2 + 1) \times 1.3 = \frac{1}{2} \times 3 \times 1.3 = 1.95 \, \text{m}^2$

Step 3: Calculate the Volume of the Pond

The volume $V$ of the pond is the area of the cross-section multiplied by the depth: $V = A \times \text{depth} = 1.95 \, \text{m}^2 \times 0.5 \, \text{m} = 0.975 \, \text{m}^3$

Since $1 \, \text{m}^3 = 1000 \, \text{liters}$, the volume of the pond is: $0.975 \times 1000 = 975 \, \text{liters}$

Step 4: Determine the Rate at Which Water is Being Pumped Out

The problem states that the water level decreases by 20 cm (or 0.2 meters) in the first 30 minutes.

Since the area of the cross-section is $1.95 \, \text{m}^2$, the volume of water removed in 30 minutes is: $\text{Volume removed} = 1.95 \, \text{m}^2 \times 0.2 \, \text{m} = 0.39 \, \text{m}^3$

Converting this volume to liters: $0.39 \times 1000 = 390 \, \text{liters}$

So, the pump removes 390 liters every 30 minutes.

Step 5: Calculate the Total Time to Empty the Pond

Now, we need to find how long it will take to remove the remaining water. We started with 975 liters, so the remaining water after the first 30 minutes is: $975 - 390 = 585 \, \text{liters}$

Since the pump removes 390 liters every 30 minutes, we can calculate the additional time required to empty the remaining 585 liters: $\text{Number of 30-minute intervals} = \frac{585}{390} \approx 1.5$

This means it will take an additional 1.5 intervals of 30 minutes each to empty the pond. Converting this to hours and minutes:

• $1$ interval of 30 minutes is 0.5 hours, so $1.5 \times 30 = 45$ minutes.

Final Answer

The additional time Sumeet has to wait is:

• 0 hours and 45 minutes.

Would you like further details, or do you have questions about any of these steps?

Related Questions:

1. How would the time change if the pond had a different depth?
2. What would be the effect on time if the pump rate changed?
3. How would you calculate the time if the shape of the pond were a different polygon?
4. What are other methods to calculate the volume of irregular shapes?
5. How would this calculation change if the water were pumped out at a varying rate?

Tip: Always check unit consistency when solving geometry-based volume problems to avoid errors in the final calculation.