Problema Solution
The revenue achieved by selling x graphing calculators is figured to be x(32-0.5x) dollars. The cost of each calculator is $24. How many graphing calculators much be sold to make a profit (revenue-cost) of at least $27.5?
Answer provided by our tutors
let
r=revenue
c=cost
p=profit
x=number of graphing calculators
given
r = x(32 - 0.5x) = 32x - 0.5x^2
c = 24x
p = r - c = 32x - 0.5x^2 - 24x = 8x - 0.5x^2
p = 8x - 0.5x^2
we need to find p, such that p>=27.50
8x - 0.5x^2 > 27.50
by simplifying we get
-x^2 + 16x - 45 =>0
x1 = (-16 + (256 - 180))/-2
x2 = (-16 - (256 - 180))/-2
x1 = 8 - 19^0.5
x2 = 8 + 19^0.5
plot the graph of the function f(x) = -x^2 + 16x - 45
you will see that for x in the interval [8 - 19^0.5, 8 + 19^0.5] we have -x^2 + 16x - 45 =>0
that is for 4 <= x <= 12 the profit p >= 27.5
We must sell between 4 and 12 calculators to make a profit of at least $27.5.