Practice Exercise Sheet 1 - Trinity College Dublin
1. Determine whether solutions exist for each of the following quadratic equations. Where they do find the solution(s).
Firstly determine whether solutions exist using the following criteria:
[pic] Two solutions
[pic] One solution
[pic] No solution
Secondly find the solution where possible using the formula:
[pic]
(i) [pic]
a=1, b=-2, c=0
[pic] two solutions exist
[pic]
[pic]
[pic]
(ii) [pic]
Multiply out the quadratic
[pic]
Divide across by 3
[pic]
a=1, b=-1, c=-2
[pic] two solutions exist
[pic]
[pic]
[pic]
(iii) [pic]
a=9, b=-24, c=16
[pic] one solution
[pic]
(iv) [pic]
a=3, b=2, c=3
[pic] no solution
(v) [pic]
a=2, b=11, c=-21
[pic] two solutions
[pic]
[pic] [pic]
(vi) [pic]
a=-2, b=1, c=10
[pic] two solutions
[pic]
[pic] [pic]
2 A firms demand function for a good is given by P = 107-2Q and their total cost function is given by TC = 200+3Q .
i) Obtain an expression for total revenue profit in terms of Q
Total Revenue = P.Q
TR = (107-2Q)*Q = 107Q-2Q2
Profit = TR-TC
Profit = 107Q-2Q2-200-3Q = -2Q2+104Q-200
ii) For what values of Q does the firm break even
Firm breaks even where Profit = 0
-2Q2+104Q-200 = 0
a = -2, b=104, c=-200
[pic]
iii) Illustrate the answer to (ii) using sketches of the total cost function, the total revenue function and the profit function
Note: Break even where Profit = 0 or TR=TC.
iv) From the graph estimate the maximum profit and the level of output for which profit is maximised
Maximum profit at max point on profit curve.
Max profit = 1150 at Q = 26
3. What is the profit maximising level of output for a firm with the marginal cost function MC = 1.6Q2-15Q+60 and a marginal revenue function MR = 280-20Q?
Profit is maximised where MR=MC
280-20Q = 1.6Q2-15Q+60
1.6Q2+5Q-220=0
a=1.6, b=5, c=-220
[pic]
Profit maximising level of output is Q = 10.27 (can’t have negative output)
4. The demand function for a good is given as Q = 130-10P. Fixed costs associated with producing that good are €60 and each unit produced costs an extra €4.
i) Obtain an expression for total revenue and total costs in terms of Q
TR = P.Q
Q = 130-10P
10P = 130-Q
P = 13-Q/10
TR = (13-Q/10)*Q = 13Q-0.1Q2
TC = FC+VC
TC = 60+4Q
ii) For what values of Q does the firm break even
Firm breaks even where TR = TC
13Q-0.1Q2=60+4Q
-0.1Q2+9Q-60=0
a=-0.1, b=9, c=-60
[pic]
iii) Obtain an expression for profit in terms of Q and sketch its graph
iv) Use the graph to confirm your answer to (ii) and to estimate maximum profit and the level of output for which profit is maximised
Profit = TR-TC
Profit = 13Q-0.1Q2-60-4Q=-0.1Q2+9Q-60
-----------------------
Profit
TR
TC
Profit
Q = 26
TR
TC
Proft = 1150
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