The time(t) taken to buy fuel at a filling station varies directly as the number of vehicles(V) in a queue and varies inversely as the number of pumps (P), available at the station. In a station with 5 pumps, it took 10 minutes to fuel 20 vehicles. find the;
(a) relationship between t, P, and V;
(b) time it takes to fuel 50 vehicles at a station with 2 pumps;
(c) number of pumps required to fuel 40 vehicles in 20 minutes.
Explanation
Given P = 5, t = 10mins, V = 20
(a) t ∝ \(\frac{\text{V}}{\text{P}}\)
t = \(\frac{\text{KV}}{\text{P}}\)
K = \(\frac{\text{tP}}{\text{V}}\) = \(\frac{10 \times 5}{20}\) = \(\frac{5}{2}\)
Therefore, the relationship between t, P, and V is t = \(\frac{\frac{5}{2}V}{\text{P}}\) = \(\frac{5V}{2P}\)
(b) to find t, when V = 50 and P = 2
t = \(\frac{5 \times 50 }{2 \times 2}\) = \(\frac{250}{4}\) = 62.5mins.
(c) to find P when V = 40 and t = 20mins
P = \(\frac{5V}{2t}\) (simply by switching t for P)
P = \(\frac{5 \times 40}{2 \times 20}\) = \(\frac{200}{40}\) = 5 pumps.