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If a firm faces the marginal cost schedule MC = 180 + 0.3 q^2 and the marginal revenue schedule is MR = 540 – 0.6 q^2 and total fixed cost is Rs....

First, we need to figure out what the profit-maximizing quantity sold would be, given these demand and supply schedules.For that, set marginal revenue equal to marginal cost:


MC = 180 + 0.3 q^2 = MR = 540 – 0.6 q^2


0.9 q^2 = 720q^2 = 800


q = 20 sqrt(2) = 28.28


Now we need to figure out what the total revenue and total cost would be.That means we need to integrate ...

First, we need to figure out what the profit-maximizing quantity sold would be, given these demand and supply schedules.

For that, set marginal revenue equal to marginal cost:


MC = 180 + 0.3 q^2 = MR = 540 – 0.6 q^2


0.9 q^2 = 720
q^2 = 800


q = 20 sqrt(2) = 28.28


Now we need to figure out what the total revenue and total cost would be.

That means we need to integrate both of these marginal functions, and then find the appropriate initial condition to set the constants.

TR = int MR dq = int 540 - 0.6 q^2 dq = 540 q - 0.2 q^3 + C
What is the constant? Well, if we sell zero things, we should get zero revenue. So C = 0.

TR = 540 q - 0.2 q^3

TC = int MC dq = int 180 + 0.3 q^2 dq = 180q + 0.1 q^3 + D
If we sell zero things, is our cost zero? No. We still pay our fixed cost. We're told that our fixed cost is in fact Rs 65. So D = 65.

TC = 180 q + 0.1 q^3 + 65

Now all we have to do is subtract these two to get profit, and then substitute in the profit-maximizing quantity we found earlier:

TP = TR - TC = 540 q - 0.2 q^3 - (180 q + 0.1 q^3 + 65)
TP = 360 q - 0.3 q^3 - 65
TP = 360 (20 sqrt(2)) - 0.3 (20 sqrt(2))^3 - 65
TP = 7,200 sqrt(2) - 4,800 sqrt(2) - 65
TP = 2,400 sqrt(2) - 65 = 3,329.11

Thus, if they properly maximize profit, their profit will be Rs 3,329.11.

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