Given the folllowing revenue and cost functions, find the maximum profit.

Question

R(x)=72x-2x²; C(x)=24x+93.

Answer 1

p = r - c
p = -2 x^2 + 48 x - 93

find the vertex of that parabola by completing the square (unless you know calculus)

x^2 - 24 x + 93/2 = -p/2

x^2 - 24 x = -p/2 -93/2

x^2 - 24 x + 144 = -p/2 - 93/2 + 288/2

(x-12)^2 = -(1/2)(p-195)

max at x = 12 and profit = 195
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Now if you know calculus,
p = -2 x^2 + 48 x - 93
dp/dx = 0 at max = -4x+48
4x = 48
x = 12
p = -2(144) +48*12 -93
p = 195