Consumption of fuel by car varies exponentially with respect to speed. When a car travels at a speed of 100 kmph, it uses 25% more petrol to travel a certain distance than it does when it travels at a speed of 75 kmph. If the mileage of the car is 25 kmpl at a speed of 75 kmph, then what can be the amount of petrol required (in liters) to travel a distance of 225 km in 150 minutes

To solve this problem, we need to calculate the amount of petrol consumed by the car to travel a given distance at different speeds, considering that the fuel consumption varies exponentially with speed.

### 1. Understanding the Problem

Given:
1. **Speed and fuel consumption relationship**: Fuel consumption varies exponentially with speed.
2. **Fuel consumption data**:
   - At 75 kmph, the mileage is 25 kmpl (kilometers per liter).
   - At 100 kmph, the car uses 25% more petrol to travel the same distance compared to 75 kmph.

### 2. Calculating Fuel Consumption at Different Speeds

Let's denote the fuel consumption rate (in liters per km) as \( C \). The mileage (kmpl) is the reciprocal of the consumption rate.

#### At 75 kmph:
- Mileage = 25 kmpl
- Consumption rate at 75 kmph, \( C_{75} \):
  \[
  C_{75} = \frac{1}{\text{Mileage at 75 kmph}} = \frac{1}{25} \text{ liters per km}
  \]

#### At 100 kmph:
- The car uses 25% more fuel to travel the same distance as compared to 75 kmph. Thus, the consumption rate at 100 kmph, \( C_{100} \):
  \[
  C_{100} = C_{75} \times 1.25 = \frac{1}{25} \times 1.25 = \frac{1.25}{25} = \frac{1}{20} \text{ liters per km}
  \]

### 3. Calculating the Distance and Speed

Given:
- Distance to travel = 225 km
- Time taken = 150 minutes

To find the speed:
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]
First, convert 150 minutes into hours:
\[
150 \text{ minutes} = \frac{150}{60} \text{ hours} = 2.5 \text{ hours}
\]
\[
\text{Speed} = \frac{225 \text{ km}}{2.5 \text{ hours}} = 90 \text{ kmph}
\]

### 4. Estimating Fuel Consumption at 90 kmph

Since fuel consumption varies exponentially with speed, and we have data for 75 kmph and 100 kmph, we need to estimate the consumption at 90 kmph. We'll use a weighted approach since 90 kmph lies between 75 kmph and 100 kmph:

- Calculate the average consumption rate as an approximation:
\[
C_{90} \approx \frac{C_{75} + C_{100}}{2} = \frac{\frac{1}{25} + \frac{1}{20}}{2}
\]
\[
C_{90} \approx \frac{\frac{4}{100} + \frac{5}{100}}{2} = \frac{9}{200} = 0.045 \text{ liters per km}
\]

### 5. Calculating the Total Fuel Required

- Total distance = 225 km
- Estimated consumption rate at 90 kmph = 0.045 liters per km

Total fuel required:
\[
\text{Total fuel} = 225 \text{ km} \times 0.045 \text{ liters per km} = 10.125 \text{ liters}
\]

### Final Answer
The amount of petrol required to travel 225 km in 150 minutes at an average speed of 90 kmph is approximately **10.125 liters**.

---

### Do you have any questions or would you like further details?

### Related Questions:

1. How would the fuel requirement change if the distance was different but the time remained constant?
2. What would be the fuel consumption at 110 kmph if the same exponential relationship holds?
3. How does the exponential variation of fuel consumption with speed impact the total cost of a trip?
4. What other factors, besides speed, can influence fuel consumption in cars?
5. Can we derive a mathematical formula for the relationship between speed and fuel consumption?

### Tip:
Understanding how external factors like speed and driving habits impact fuel consumption can help optimize fuel efficiency and reduce costs.

Explore how petrol consumption varies with speed using exponential functions. Calculate the amount of petrol required to travel 225 km in 150 minutes at an average speed of 90 kmph. Includes step-by-step solution and formulas for exponential growth and average rate of change. Ideal for students and enthusiasts studying exponential functions and their applications.

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