Optimization Question about Profit, Demand, Revenue

Question

I have a function of R(x) and C(x), which are income earned/Revenue function and cost function per week respectively.

For example, let's say:

R(x) = 310x + 0.02x^2 - 0.003x^3

C(x) = 3400 - 23x + 0.05x^2 + 0.0004x^3

I also have a profit function, that is P(x) = R(x) - C(x)

The main question is that if the manufacturing is increasing to 52 smartphones per week, and their current output is 130 smartphones, what is the rate that their profit will be increasing?

Thank you very much!

Answer 1

I'm not certain exactly what the problem is asking, but for what it's worth here are some somewhat relevant calculations

R(x) = 310 + .02x^2 - .003x^3

C(x) = 3400 - 23x + .05x^2 + .0004x^3

P(x) = R(x) -C(x) = (.003-.0004)x^3 + (.02-.05)x^2 +23x + (310-3400)

take the derivative to find the rate of change in profit at production level of 130+54 = 184

P(x) = .0026x^3 -.03x^2 + 23x -3090

P'(x) = .0078x^2 -.06x +23

P'(184) = .0078(184)^2 - .06(184) + 23

= 264.0768 - 11.04+ 23

= 298.1168= $298.12 increase

P(130) = .0026(130)^3 -.03(130)^2 +23(130) -390

= 5712,2 - 507 + 2990 -390

= 7805.2

P(130+54) = .0026(184)^3 -.03(184)^2 +23(184) -390

= 16196.7104- 1015.68 + 4232 -390

= 19,023.030304

Profit increased by 11,217.8304 from 7,805.2 to 19,023.030304

298.12/19,023.030304 = about 2% per phone

which is a 11,217.8304/7,805.2 = 143,7225234%= about 144% increase in profit from 130 to 184

144/54 = about 2.7% rate of increase per cell phone

P(184)-P(130) = 11,217.8304

[P(184)-P(130)]/54 = 207.7376

P(184+54) = P(238) = .0026(238)^3 -.03(238)^2 +23(238) -390

= 35051.3072 - 1699.32 + 5474 -390

= 38,435.9872

P(238)- P(184) = 19,412.9569

[P(238) - P(184)]/54 = 359.4992019

(207.7376 + 359.4992019)/2 = 283.6184009

which equals close to P'(184) = 298.1168 =rate change at a point in time when x=130+54

283.6184009 = rate of change over an interval of x going from 130 to 130+54+54, with midpoint 103+54

if you found a smaller interval, plus & minuse 184, it should approach 298.1168

the answer you probably want is 298.1168 = P'(130+54) = the derivative of the profit function when x increases from 130 to 130+54

which is about a 144% increase in profits from x=130 to x=130+54, which is an average 144/54 = 2.67% increase in profit per cell phone

(no guarantees, arithmetic is error free)