Practice Exercise Sheet 1

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:

Two solutions
One solution
No solution

Secondly find the solution where possible using the formula:

(i)
a=1, b=-2, c=0
two solutions exist



(ii)
Multiply out the quadratic

Divide across by 3

a=1, b=-1, c=-2
two solutions exist





(iii)
a=9, b=-24, c=16
one solution



(iv)
a=3, b=2, c=3
no solution


(v)
a=2, b=11, c=-21
two solutions

(vi)

a=-2, b=1, c=10
two solutions

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 .

1. 
   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

2. 
   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

3. 
   I
   Proft = 1150
   llustrate the answer to (ii) using sketches of the total cost function, the total revenue function and the profit function
   

N
Profit

TR

TC


Profit

TR

TC

Q = 26
ote: Break even where Profit = 0 or TR=TC.

4. 
   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

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.

1. 
   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

2. 
   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

3. 
   Obtain an expression for profit in terms of Q and sketch its graph
   

4. 
   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