Theory question 108
Logarithms(a) Solve the simultaneous equations:
log10x + log10y = 4
log10x + 2 log10y = 3
(b) The time t taken to buy fuel at a petrol station varies directly as the number of vehicles V on queue and jointly varies inversely as the number of pumps P available in the station. In a station with 5 pumps, it took 10 minutes to fuel 20 vehicles. Find (i) the relationship between t, P and V; (ii) the time it will take to fuel 50 vehicles in the station with 2 pumps; (iii) the number of pumps required to fuel 40 vehicles in 20 minutes.
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(a) Subtract the first equation from the second: log10y = −1, so y = 1/10.
Then log10x = 4 − (−1) = 5, so x = 100,000.
(b)(i) t = kV/P. 10 = k × 20/5 gives k = 5/2, so t = 5V/(2P).
(ii) t = 5 × 50/(2 × 2) = 62.5 minutes (1 hr 2½ min).
(iii) 20 = 5 × 40/(2P), so P = 200/40 = 5 pumps.
Treat log x and log y as the unknowns and eliminate one of them.