This is a project on elementary calculus; it shows the production cost per edition that include total revenue function, profit function, marginal revenue function, marginal cost function, marginal profit function and average cost function. It also explains the manipulation of the total revenue, profit, marginal revenue, marginal cost, marginal profit and average cost.
¶ … Production Cost Per Edition Is
TC (Q) =70+0.10Q+0.001Q2
Functions
(i) Total Revenue Function
Total Revenue is normally calculated by multiplying the price of the product with the quantity sold.
TR (Q) =P (Q) x Q. where Q. is the quantity of output sold, and P (Q) is the inverse demand function of the price.
Price per unit is
Simplified function 0.90Q2
Profit Function
The profit is calculated by subtracting the production cost from the total revenue
(Q) = TR (Q) - TC (Q)
Production cost =70+0.10Q+0.001Q2
P (Q) x Q -- (70+0.10Q+0.001Q2) where P. Or ? is the price per Unit
PQ2-70+0.10Q+0.001Q2
Price per copy is
Q2 -70+0.10Q+0.001Q2
Profit Function = 0.9010 Q2 +0.10Q -70
Marginal Revenue Function
The marginal revenue is the extra revenue that comes from selling 1 additional unit. The change in revenue with respect to a change in quantity must be computed first.
MR (Q)= d (TR) Where'd represent derivative dQ
Total Revenue function = 0.90Q2
MR (Q)= d (TR) =d (0.90Q2)
dQ d (Q)
Marginal cost Function
The marginal unit is the last unit. Therefore, marginal cost is the cost of the last unit, or what it costs to produce one more unit. The change in costs from a previous level divided by the change in quantity from the previous level is represented by MC = Change in TC / Change in Q
MC (Q)= d (TC)
d (Q)
The function is =MC (Q)=d (70+0.10Q+0.001Q2)
d (Q)
Marginal profit Function
M ? = R ' (Q) - C ' (Q)
Marginal revenue is d (0.90Q2) - d (70+0.10Q+0.001Q2)
d (Q) d (Q)
M ? = 0.90Q2 - (70+0.10Q+0.001Q2 )
M ? = 0.90Q - (70+0.10Q+0.001Q2 )
Q
Average cost function
Average cost is total revenue divided by the number of quantity
MC= 0.90Q2
Q
Manipulate the total revenue, profit, marginal revenue, marginal cost, marginal profit and average cost given that Q=500
Total revenue
TR (Q) =P (Q) x Q= 0.90Q2
0.90 x 5002 =0.90 x 250000
TR=225000
(i) Profit
Profit function is 0.9010 Q2 +0.10Q -70
Q=500
(0.9010 x 5002 ) + (0.10 x 500) -- 70
225250 + 50-70
P =225230
(ii) Marginal Revenue
Marginal revenue function is MR (Q) = d (TR) =d (0.90Q2)
dQ d (Q)
Q=500
=d (0.90Q2)
d (Q)
500 = 0.90 x 250000 = 450
MR = 450
(iii) Marginal cost
The MC function is d (70+0.10Q+0.001Q2)
d (Q)
Q=500
d (70+0.10Q+0.001Q2)
d (Q) = 70+0.10Q+0.001Q2
Q
70+ (0.10 x 500) + (0.001 x 5002 ) = 70+50 +250
MC = 370
(iv) Marginal profit
The marginal profit function is
M ? = 0.90Q - (70+0.10Q+0.001Q2)
Q
Q= 500
Therefore, M ? is (0.90 x 500) - 70+ (0.10 x 500) + (0.001 x 5002 )
=450-70 + 50 +250
M ? = 450-370/500 = 449.26
(v) The average cost
Average cost function is MC= 0.90Q2
Q
0.90 x 5002 = 225000/500 = 450
(b) Profit maximization
Since marginal cost of $370 is less than marginal revenue of $450, the newspaper sales should be increased to optimize profit until marginal cost equals marginal revenue.
(c) Graph of MR and MC
The point of intersection is 300
The intersection point of MR and MC meet is 300.
When the industry is perfectly competitive, the manufacturing company should face a demand curve that is identical to its marginal revenue curve (MR).Accordingly, the marginal revenue curve would have a negative gradient, due to the overall market demand curve.
(d) The production level that would make it necessary to stop producing newspaper.
When the total cost of production (TC) appears to be greater than the Total Revenue (TR), losses are realized without delay. This kind of sale creates a realized and recognized loss at the same time immediately after the sales. At this point, it would be necessary to stop the production of newspapers because the main purpose of business is to pay back the production expenses incurred and make profit at the same time.
2. Manufacturing cost of an Item is TC (Q) =6000+5Q+ Q2 / 60
(a) Average cost
Q is 1
TC = 6000+ (5 x 1) + (12/60)
TC = 6000 + 5 + 0.02
TC= 6005.02
Average cost = TC/Q
6005.02/1
AC= 6005.02
b) Average cost Graph
c. Units to be produced to ensure minimum cost
The minimum average cost quantity is where the MC is equal to AC
MC = Change in TC / Change in Q
MC = d (6005.02)
d (1)
MC= 6005.02Q
AC = 6005.02
MC=AC
6005.02Q= 6005.02
Q=1
E 4. Daily cost of producing Q. Items is given by TC (Q) =210Q+ 7000
(a) Expression for the profit function
Profit = TR -- TCs
TR= Q x Price per Unit
Price = 350
TR= 350 x Q= 350Q
(Q)=350Q - 210Q+ 7000
(Q)=140Q+7000
Profit Function = 140Q+7000
(b) Profit maximization
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