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OK HERE ARE THE CORRECTIONS. IN PROBLEM 2 THE ANSWER TO #1 IS $2 PER GALLON OF GAS. HE SAID USING THIS YOU COULD SOLVE THE REST OF THE PROBLEMS. I HOPE!!!

PLEASE SEE ATTCHMENT

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1.How much will this firm sell in each of the two markets?

Since the output from both plants can be sold in either market, the Qi on the demand side is different from Qi level of output in plant i, we define that:
In market 1, P = 484 - 20Qa; and in Market 2: P = 484 - 5Qb,
while the labels on the supply side doesn't change.

In market 1, P = 484 - 20Qa
Total Revenue 1 = P1*Qa = Qa (484 - 20Qa) = 484 Qa - 20Qa2
MR1 = dTR1 / dQa = 484- 40Qa

In market 2, P = 484 - 5Qb
Total Revenue 2 = P2*Qb = Qa (484 - 5Qb) = 484 Qb - 5Qb2
MR2 = dTR2 / dQb = 484- 10Qb

Since the firm is to maximizing its profit, it will sell the product to the two markets at a level where MR1 = MR2. Because otherwise, the firm will sell more goods to the market with higher marginal revenue.
So we set: MR1 = MR2.
484- 40Qa= 484- 10Qb
Qb = 4Qa
So the total quantity demanded is Qa + Qb = Qa + 4Qa = 5Qa

In plant 1, TC1 = 961 - 10 Q1 + Q12
Marginal cost is MC1 = 2Q1-10

Plant 2: TC2 = 324 - 10 Q2 +3Q22
Marginal cost is MC2 = 6Q2-10

Since the firm is to minimizing its cost, it will produce at a level where MC1 = MC2. Because otherwise, the firm will produce more goods in the plant with lower marginal cost.
So we set: MC1 = MC2.
2Q1-10=6Q2-10
Q1 = 3Q2
So the total quantity demanded is Q1 +Q2=3Q2 + Q2 = 4Q2

Since in the firm the total output = total sales volume
Then 5Qa = 4Q2
So Q2 = 5/4 Qa
And Q1 = 3Q2 = 15/4 Qa

Now the firm's total profit form is:
Profit = TR1 +TR2 - TC1 - TC2
= 484 Qa - 20Qa2 + 484 Qb- 5Qb2 - (961 - 10 Q1 + Q12) - (324 - 10 Q2 +3Q22 )
= 484 Qa - 20Qa2 + 484 (4Qa) - 5(4Qa) 2 - (961 - 10 (15/4 Qa) + (15/4 Qa) 2) - (324 - 10 (5/4 Qa) +3(5/4 Qa) 2 )
= 484 Qa - 20Qa2 + 1936 Qa - 20Qa 2 - 961 + 75/2 Qa - 125/16 Qa 2 -324 + 25/2 Qa +75/16 Qa 2
= -105/2 Qa 2 + 2470Qa -1285

Then the marginal profit w.r.t. Qa is dProfit / dQa = -105Qa + 2470 = 0
Solve for Qa = 23.52, which should be sold in market 1;
Then Qb = ...

$2.19