The Picture It frame company determines that the cost function for its new make-at-Home picture frame is: C(x)= 60x + 72,000, where x is the number of frames produced and C(x) is the cost in dollars. The revenue function for the same freame is: R(x)=(-x^2/30)+200x. Deterimine the profit function, P(x) and the marginal profit function, MP(x). Evaluate and interpret MP(500).
if n = 1,2,3,4... how do you get to 5,1,5,1...
I'm sorry I don't follow you..?
To determine the profit function, subtract the cost function, C(x), from the revenue function, R(x). Mathematically, the profit function, P(x), can be calculated as:
P(x) = R(x) - C(x)
Substituting the given revenue and cost functions:
P(x) = (-x^2/30) + 200x - (60x + 72,000)
Simplifying:
P(x) = -x^2/30 + 200x - 60x - 72,000
P(x) = -x^2/30 + 140x - 72,000
Now, to find the marginal profit function, MP(x), take the derivative of the profit function, P(x), with respect to x. This will give you the rate of change of profit with respect to the number of picture frames produced.
MP(x) = dP(x)/dx
Differentiating the profit function:
MP(x) = -2x/30 + 140
Evaluate MP(500) by substituting x = 500 into the marginal profit function:
MP(500) = -2(500)/30 + 140
MP(500) = -100/30 + 140
MP(500) = -10/3 + 140
MP(500) ≈ 125.33
Interpretation: MP(500) represents the rate of change of profit when 500 picture frames are produced. In this case, the marginal profit is approximately $125.33 per frame. This means that, for every additional frame produced after reaching 500, the profit of the company would increase by approximately $125.33.