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Calculus and applications to business and economics










































Hi, please help me with his maths problem.

The demand equation for Product X is given by

P = 900/q - 0.48q + 100 q > 0
Use derivaties to explore the relationship between demand for Product X,total revenue and elasticity.

Task
1. Find an expression for the total revenue,TR.
2. Find an expression for marginal revenue,MR.
3. Interpret the marginal revenue when q = 60.
4. What price must be charged to achieve a demand of q = 60.
5. Find an expression for dp/dq ,and evaluate at q = 60.
6. Use the relationship
dq/dp = 1/dp/dq , and the results of question (4) & (5), to determine wheather the demand is elastic,unit elastic or inelastic when q = 60 and interpret results.

7. Determine the value of q which maximises total revenue.
8. What price must be charged to maximise total revenue and verify that the demand is unit elastic at
this price.



Hints:
A. Do not attempt to obtain a equation for dq/dp in terms of p.

B. A second derivative is required in question (7)to varify a maximum.

C. Graph TR to verify algebraic answers.

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May 10, 2012
Calculus and applications to business and economics
by: Staff


Part I


Question:

Hi, please help me with his maths problem.

The demand equation for Product X is given by

P = 900/q - 0.48q + 100 q > 0
Use derivatives to explore the relationship between demand for Product X,total revenue and elasticity.


Task

1. Find an expression for the total revenue,TR.
2. Find an expression for marginal revenue,MR.
3. Interpret the marginal revenue when q = 60.
4. What price must be charged to achieve a demand of q = 60.
5. Find an expression for dp/dq ,and evaluate at q = 60.
6. Use the relationship
dq/dp = 1/dp/dq , and the results of question (4) & (5), to determine whether the demand is elastic, unit elastic or inelastic when q = 60 and interpret results.

7. Determine the value of q which maximizes total revenue.
8. What price must be charged to maximize total revenue and verify that the demand is unit elastic at this price.



Hints:

A. Do not attempt to obtain a equation for dq/dp in terms of p.

B. A second derivative is required in question (7)to verify a maximum.

C. Graph TR to verify algebraic answers.

Answer:

Q = quantity
Q* = Equilibrium quantity


P = price per unit
P* = equilibrium price


TR = total revenue

MR = marginal revenue

E_p_q = Price elasticity of demand = elasticity of q with respect to p

E_p_q = (% change in the quantity demand) / (% change in the price charged)






1. Find an expression for the total revenue,TR.

TR = Pq

P * q = (900/q - 0.48q + 100 q ) * q

Pq = (900/q)*q – (0.48q)*q + (100q) * q

Pq = 900*(q / q) – 0.48*q*q + 100*q*q

Pq = 900*(1) – 0.48*q*q + 100*q*q

Pq = 900 – 0.48q² + 100q²

Pq = 100q² – 0.48q² + 900

Pq = 99.52q² + 900

TR = 99.52q² + 900


2. Find an expression for marginal revenue,MR.


MR = d(TR)/dQ

d(TR)/dQ = d(99.52q² + 900)/dQ


d(TR)/dQ = 2*99.52q + 0

d(TR)/dQ = 199.04q

MR = 199.04q


3. Interpret the marginal revenue when q = 60.

MR = 199.04q

MR = 199.04*60

MR = 11942.4


4. What price must be charged to achieve a demand of q = 60.


P = 900/q - 0.48q + 100 q

P = 900/60 - 0.48*60 + 100*60

P = 15 - 28.8 + 6000

P = 5986.2


5. Find an expression for dp/dq ,and evaluate at q = 60.

P = 900/q - 0.48q + 100 q

dp/dq = 99.52 - 900/q²


if q = 60

dp/dq = 99.52 - 900/60²


dp/dq = 99.52 - 900/3600

dp/dq = 99.52 - 0.25

if q = 60, dp/dq = 99.27




May 10, 2012
Calculus and applications to business and economics
by: Staff


-----------------------------------------------------------------

Part II

6. Use the relationship

dq/dp = 1/(dp/dq) , and the results of question (4) & (5), to determine whether the demand is elastic, unit elastic or inelastic when q = 60 and interpret results.

Elasticity: the percentage change in one variable compared to the percentage change in another

E_p_q = Price elasticity of demand = elasticity of q with respect to p

E_p_q = (% change in the quantity demand) / (% change in the price charged)


Elastic: demand is elastic if the elasticity is greater than 1. When E_p_q > 1 the % change in demand is greater than the % change in price. A completely elastic demand means the elasticity is infinity (E_p_q = ∞)


Unit-elastic: demand is unit-elastic if the elasticity is 1. When E_p_q = 1 the % change in demand is equal to the % change in price.


Inelastic: the elasticity is between 0 and 1. When E_p_q < 1 the % change in demand is less than the % change in price. A completely inelastic demand means the elasticity is 0 (E_p_q = 0)


Computing the Price elasticity of demand:


Price elasticity of demand: = (dq / dp)*(P/Q)

Price elasticity of demand: = [1/(dp/dq)]*(P/Q)



when q = 60:

dp/dq = 99.27 (from question 5)

P = 5986.2 (from question 4)

Q = 60


Price elasticity of demand: = [1/(99.27)]*( 5986.2 /60)

Price elasticity of demand: = 1.0050367684094

Price elasticity of demand: = “Elastic”


7. Determine the value of q which maximizes total revenue.


Revenue increases as long as q increases.

(A second derivative will allow you to calculate the max/min of marginal revenue (MR) with respect to q)



Open the following link to view a graph of Total Revenue vs Demand (Q)


(1) If your browser is Firefox, click the following link to VIEW the graph; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

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http://www.solving-math-problems.com/images/total-revenue-calculaus-applications-2012-05-09.png


8. What price must be charged to maximize total revenue and verify that the demand is unit elastic at this price.



Price elasticity of demand: = (dq / dp)*(P/Q)

Price elasticity of demand: = [1/(dp/dq)]*(P/Q)



when q = 60:

dp/dq = 99.27 (from question 5)

P

Q = 60


when q = 60:

dp/dq = 99.27 (from question 5)

P = 5986.2 (from question 4)

Q = 60

Price elasticity of demand = 1

1 = [1/(99.27)]*( P /60)

P = 5956.20



Thanks for writing.

Staff
www.solving-math-problems.com



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