By an engine the consumption of coal is directly pro portional to the square of its velocity. When velocity is 50 mile/hour then consumption is 100 kg. If cost of coal is 25paise /kg, and expenditure is 9 Rs./hour in second case. Then find the consumption in travelling 250 mile.

Options

A. 325 kg.
B. 300 kg.
C. 225 kg.
D. 270 kg.

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the relationship between coal consumption and velocity
The problem states that the consumption of coal (C) is directly proportional to the square of its velocity (V). This can be expressed mathematically as:
\[ C = k \cdot V^2 \]
where \( k \) is the constant of proportionality.

### Step 2: Find the constant of proportionality (k)
We know from the problem that when the velocity is 50 miles/hour, the consumption is 100 kg. We can substitute these values into the equation to find \( k \):
\[ 100 = k \cdot (50)^2 \]
\[ 100 = k \cdot 2500 \]
\[ k = \frac{100}{2500} = \frac{1}{25} \]

### Step 3: Determine the consumption in the second case
We are given that the expenditure is 9 Rs/hour and the cost of coal is 25 paise/kg. First, we convert the cost of coal to Rs:
\[ \text{Cost of coal} = \frac{25}{100} = 0.25 \text{ Rs/kg} \]

Now, we can find the consumption of coal in the second case using the expenditure:
\[ \text{Consumption} = \frac{\text{Expenditure}}{\text{Cost per kg}} \]
\[ \text{Consumption} = \frac{9 \text{ Rs/hour}}{0.25 \text{ Rs/kg}} = 36 \text{ kg/hour} \]

### Step 4: Set up the equation for the second case
Using the same formula for coal consumption:
\[ C_2 = k \cdot V_2^2 \]
We know \( C_2 = 36 \) kg and \( k = \frac{1}{25} \):
\[ 36 = \frac{1}{25} \cdot V_2^2 \]
Multiplying both sides by 25:
\[ 900 = V_2^2 \]
Taking the square root:
\[ V_2 = 30 \text{ miles/hour} \]

### Step 5: Calculate the time to travel 250 miles
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Substituting the values:
\[ \text{Time} = \frac{250 \text{ miles}}{30 \text{ miles/hour}} = \frac{250}{30} = \frac{25}{3} \text{ hours} \]

### Step 6: Calculate the total coal consumption for the trip
Now we can find out how much coal is consumed during this time:
\[ \text{Total Consumption} = \text{Consumption per hour} \times \text{Time} \]
\[ \text{Total Consumption} = 36 \text{ kg/hour} \times \frac{25}{3} \text{ hours} \]
\[ \text{Total Consumption} = 36 \times \frac{25}{3} = 300 \text{ kg} \]

### Final Answer
The total consumption of coal when traveling 250 miles is **300 kg**.

---