A telecommunications company has 50,000 units of mobile cellular subscribers and charges P8.00 per minute of voice calls. The officials believe that if the charge is reduced, the number of subscribers will increase at the rate of 1000 units for each centavo reduction. What charge will yield the gross revenue on this basis?
I assume these are Philippine pesos and there are 100 centavos in a peso.
current subscribers --- 50,000
cost per minute = 800 centavos
let the number of one centavo reductions be n
subscribers = 50000 + 1000n
cost per minute = 800 - n
Revenue
= (50000+1000n)(800-n)
= 1000(50 + n)(800-n)
= 1000(40000 + 750n - n^2)
-n^2 + 750n + 40000 is quadratic, and if y = -n^2 + 750n + 40000
we have a parabola which opens downwards and the n of the vertex is
-750/-2 = 375
so the charge per minute to yield the greatest revenue is 800-375
or 425 centavos or P4.25
I took the easiest way to find the vertex
you could also complete the square
or use Calculus,
d(-n^2 + 750n + 40000)/dn
= -2x + 750
= 0 for a max
2n=750
n = 375