Page:Calculus Made Easy.pdf/137

 Now

which is positive for all the values of $$P$$; hence $$P = +\sqrt{\dfrac{b}{a}} - c$$ corresponds to a minimum.

(5) The total cost per hour $$C$$ of lighting a building with $$N$$ lamps of a certain kind is

where $$E$$ is the commercial efficiency (watts per candle),
 * $$P$$ is the candle power of each lamp,
 * $$t$$ is the average life of each lamp in hours,
 * $$C_l=$$ cost of renewal in pence per hour of use,
 * $$C_e=$$ cost of energy per $$1000$$ watts per hour.

Moreover, the relation connecting the average life of a lamp with the commercial efficiency at which it is run is approximately $$t=mE^n$$, where $$m$$ and $$n$$ are constants depending on the kind of lamp.

Find the commercial efficiency for which the total cost of lighting will be least.

for maximum or minimum.