Question 1175979: The profit P per day from selling x units of a commodity is given by P=x(200-0.05x). How many units of the commodity must be sold in order to attain the daily maximum profit? What is the daily maximum profit?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
P(x) = x(200-0.05x)
P(x) = 200x-0.05x^2
P(x) = -0.05x^2 + 200x + 0
y = -0.05x^2 + 200x + 0
The last equation is in the form y = ax^2+bx+c
where,
a = -0.05
b = 200
c = 0
The parabola opens downward because the leading coefficient a = -0.05 is negative.
This means the vertex is the highest point where the max profit occurs.
The vertex is (h,k) such that
h = -b/(2a)
h = -200/(2(-0.05))
h = 2000
and
k = P(h)
k = -0.05h^2 + 200h + 0
k = -0.05(2000)^2 + 200(2000) + 0
k = 200,000
If you sell x = 2000 units per day, then you'll reach the max profit of P = 200,000 dollars per day.
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