Question 2 The time ( \( t \) ) taken to buy a fuel at a petrol station Variation varies directly as The number of vechice \( v \) on a quenc and inversely as the number of pumps (p) available in a station. In a station of 5 pumps, it took 10 minutes to fuel 20 vechiles Find; (a) The relationship betweet \( t \) \( p \) and \( U \). (b) The time it will tablie to puel 50 vechiles in station usitte 2 pumps

To tackle this problem, we first need to derive the relationship. Since time \( t \) varies directly as the number of vehicles \( v \) and inversely as the number of pumps \( p \), we can express it as: \[ t = k \cdot \frac{v}{p} \] Where \( k \) is a constant. Given that it took 10 minutes to fuel 20 vehicles at a station with 5 pumps, we can substitute these values to find \( k \): \[ 10 = k \cdot \frac{20}{5} \implies k = 10 \cdot \frac{5}{20} = 2.5 \] Now we have: \[ t = 2.5 \cdot \frac{v}{p} \] To find the time it will take to fuel 50 vehicles at a station with 2 pumps, we substitute \( v = 50 \) and \( p = 2 \): \[ t = 2.5 \cdot \frac{50}{2} = 2.5 \cdot 25 = 62.5 \text{ minutes} \] So, it will take 62.5 minutes to fuel 50 vehicles with 2 pumps. Did you know that in some countries, the number of vehicles at fuel stations can increase significantly during holiday seasons or major events, leading to longer waiting times? Knowing how to optimize the number of pumps and manage queues can significantly impact customer satisfaction. Moreover, while calculating relationships like this, common mistakes include forgetting to correctly apply direct and inverse relationships or misplacing constants. Double-checking each step can prevent such errors and help you nail those math problems!