Suppose the total cost in dollars per week by Capital Corporation for producing its best-selling product is given by 𝐶(𝑥) = 1000 + 300𝑥. The revenue function for the production of the product was derived to be 𝑅(𝑥) = 800𝑥 − 0.6𝑥2. Analyse the profit function for the manufacturing of 𝑥 products, as well as interpret what the marginal cost and marginal revenue of 15 products signifies. Additionally, determine the exact and approximate profit from the manufacturing of the 42nd product.
P(x) = R(x) - C(x) = -1000 + 500 x - 0.6 x^2
dP/dx = 500 - 1.2 x
d^2P/dx^2 is -1.2 for all x so max when dP/dx = 0
max profit when x = 500/1.2 = 417
if x = 15
R = 800*15 - 0.6*225 = 12000-135 = 11,865
C = 1000 + 300 (15) = 1000 + 4500 = 5,500
P = R-C = 6,365
dP/dx = 500 - 1.2*15 = 482
dR/dx = 800 - .6*225 = 800 - 135 = 665
dC/dx = 300 always
etc, you can do for 42
marginal cost/revenue = dC/dx or dR/dx
profit p(x) = R(x)-C(x) = -0.6x^2 + 500x - 1000
the marginal profit is thus dp/dx = -1.2x+500
for the 42nd unit, that would be 449.6
the exact profit would be p(42)-p(41) = 450.2
To analyze the profit function, we first need to understand what it represents. The profit function measures the difference between the revenue earned and the cost incurred in producing a given number of products.
The profit function, denoted as P(x), can be calculated by subtracting the cost function (C(x)) from the revenue function (R(x)):
P(x) = R(x) - C(x)
In this case, the revenue function is given as R(x) = 800x - 0.6x^2, and the cost function is given as C(x) = 1000 + 300x.
To find the profit function for manufacturing x products, substitute R(x) and C(x) into the profit function formula:
P(x) = (800x - 0.6x^2) - (1000 + 300x)
Simplifying the equation further:
P(x) = 800x - 0.6x^2 - 1000 - 300x
P(x) = -0.6x^2 + 500x - 1000
To interpret the marginal cost and marginal revenue of 15 products, we need to find their respective derivatives. The derivative of the cost function represents the marginal cost (MC), while the derivative of the revenue function represents the marginal revenue (MR).
The marginal cost (MC) can be calculated by taking the derivative of the cost function C(x):
MC(x) = d/dx [C(x)]
MC(x) = d/dx [1000 + 300x]
MC(x) = 300
The marginal cost (MC) is a constant 300, meaning the cost of producing each additional unit remains constant at $300.
The marginal revenue (MR) can be calculated by taking the derivative of the revenue function R(x):
MR(x) = d/dx [R(x)]
MR(x) = d/dx [800x - 0.6x^2]
MR(x) = 800 - 1.2x
To determine the marginal cost and marginal revenue of 15 products, substitute x = 15 into their respective formulas:
MC(15) = 300
MR(15) = 800 - 1.2(15) = 780
The marginal cost of producing 15 products is $300, indicating that the cost of producing an additional unit remains constant at $300.
The marginal revenue from the sale of the 15th product is $780, meaning that the revenue generated by selling one additional unit is $780.
To find the exact profit from manufacturing the 42nd product, substitute x = 42 into the profit function P(x):
P(42) = -0.6(42)^2 + 500(42) - 1000
P(42) = -0.6(1764) + 21000 - 1000
P(42) = -1058.4 + 21000 - 1000
P(42) = 19841.6 - 1000
P(42) = $18,841.6
Therefore, the exact profit from manufacturing the 42nd product is $18,841.6.
To approximate the profit from manufacturing the 42nd product, we use the profit function P(x) and substitute x = 42:
P(42) = -0.6(42)^2 + 500(42) - 1000
P(42) ≈ -0.6(1764) + 21000 - 1000
P(42) ≈ -1058.4 + 21000 - 1000
P(42) ≈ 19841.6 - 1000
P(42) ≈ $18,841.6
Therefore, the approximate profit from manufacturing the 42nd product is also $18,841.6.