The Apex Widget Company has determined that its marginal cost function is C\'(x)
ID: 1208125 • Letter: T
Question
The Apex Widget Company has determined that its marginal cost function is C'(x) = 0.5x + 50, where x represents the number of boxes of widgets which can be produced each week and C is in dollars. The cost of producing 200 boxes of widgets is $22,500. Also, if they charge $50 per box, they will sell 350 boxes; and if they charge $75 per box they will sell 300 boxes. Assume this demand equation is linear.
1. Suppose that plant capacity is such that not more than 200 boxes of widgets can be produced each week. Find the following answers below.
(a) the demand equation
(b) the revenue function
(c) the number of boxes which they should sell to maximize revenue.
(d) the maximum revenue.
(e) the total cost function
(f) the average cost function.
(g) the production level (number of boxes) which yields minimum average cost.
(h) the minimum average cost.
(i) the profit fuction.
(j) the number of boxes which they should produce and sell to maximize profit.
(k) the maximum profit.
(l) the price which should be charged in order to maximize the profit.
Explanation / Answer
Ans:
a)
D(50) = 350
D(75) = 300
Direct (y = deal cost)
D(y) = Ay + b
A = - 2
b = 450
D(y) = 450 - 2y
b) revenue function
Income would be yD(y)
R(y) = 450y - 2y^2
R'(y) = 450 - 4y
R'(y) = 0 @ 112.5
Max income @ 112.5 boxes
Max income + 450(112.5) - 2(112.5)^2
Max income = 25,313
So if C'(x) = 0.5x + 50 then you coordinate to get C(x) which is the thing that you truly think about.
C(x) = 0.25x^2 + 50x + c
C(200) = 22,500 = 0.25(200)^2 + 50(200) + c
c = 2500
c)
C(x) = 0.25x^2 + 50x + 2500
Normal expense would be C(x)/x
AvgC(x) = 0.25x + 50 + 2500/x
Normal expense has no worldwide least, it diminishes consistently as generation increments.
d)
Benefit is income - cost
P(y) = 450y - 2y^2 - (C(D(y))
P(y) = 450y - 2y^2 - [0.25(450-2y)^2 + 50(450-2y) + 2500] which lessens to
P(y) = - 3y^2 + 1000y - 75625
e)
P'(y) = - 6y + 1000 which is zero @ y = 167
As characterized from above, y = deal cost
f)
D(167) = 450 - 2(167) = 116 boxes
Max Profit = - 3(167)^2 + 1000(167) - 75625
Max Profit = $7708 @ $167/box
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