  # Calculus and applications to business and economics

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.

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.

### Comments for Calculus and applications to business and economics

 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