Assume that the rate of consumption of coal by a locomotive varies as the square of the speed and is 1000 kg per hour when the speed is 60 km per hour. If the coal costs the railwa... Assume that the rate of consumption of coal by a locomotive varies as the square of the speed and is 1000 kg per hour when the speed is 60 km per hour. If the coal costs the railway company ₹15 per 100 kg and if the other expenses of the train be ₹12 per hour, find a formula for the cost in paise per kilometer when the speed is S km per hour. Select the appropriate answer from the options given below. Read more

Steps to Solve

1. Understand the coal consumption relation
   According to the problem, the rate of coal consumption varies as the square of the speed. We are given that at a speed of 60 km/h, the consumption is 1000 kg.

2. Set up the equation for coal consumption
   If we denote speed as $S$ (in km/h), then the coal consumption can be expressed as: $$ C = k \cdot S^2 $$ where $k$ is a constant.

3. Find the constant $k$
   Using the known values:
   At $S = 60$ km/h, $C = 1000$ kg, we can substitute into the equation: $$ 1000 = k \cdot (60)^2 $$ This gives us: $$ k = \frac{1000}{3600} = \frac{5}{18} $$

4. Write the coal consumption formula
   Now, substituting $k$ back into the equation gives us: $$ C = \frac{5}{18} S^2 $$

5. Calculate the cost of coal
   The cost of coal per 100 kg is ₹15, so the cost per kg is: $$ \text{Cost per kg} = \frac{15}{100} = 0.15 \text{₹/kg} $$
   Thus, the total cost for $C$ kg of coal is: $$ \text{Total coal cost} = 0.15 \cdot C = 0.15 \cdot \frac{5}{18} S^2 $$

6. Combine costs to find total cost
   Including the other expenses (₹12 per hour), the total cost becomes: $$ \text{Total cost} = \frac{0.15 \cdot \frac{5}{18} S^2 + 12}{S} $$

7. Convert to paise and simplify
   Since 1 ₹ = 100 paise, multiplying the entire expression by 100 yields: $$ \text{Total cost in paise} = \frac{100 \left( 0.15 \cdot \frac{5}{18} S^2 + 12 \right)}{S} = \frac{75 S^2 + 1200}{S} $$