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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