   Chapter 11.5, Problem 24E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 15-24, p is the price per unit in dollars and q is the number of units.If the demand and supply functions for a product are p   = 5000   −   20 q  -  0.7 q 2  and  p   = 500   +   10 q   +   0.3 q 2 , respectively, find the tax per unit t that will maximize the tax revenue T.

To determine

To calculate: The tax per item that will maximize the total tax revenue if the demand function for a fixed period of time is given by p=500020q0.7q2 and the supply function before taxation is p=(0.3)q21+10q+500.

Explanation

Given Information:

The provided expression is the demand function for a fixed period of time is given by p=500020q0.7q2 and the supply function before taxation is p=(0.3)q21+10q+500.

If the supply function is f(q), then after taxation, the new supply function is equal to f(q)+t.

Where t is the tax per unit.

The total tax T is equal to T=tq, where t is the tax per unit and q is the number of units.

The tax revenue is maximized by finding the value of the variable at first derivative is equal to 0. Also checking by verifying that T(q)0 or not.

Formula Used:

The following procedure are used to maximizing total tax revenue,

Step-1 Write the supply function after taxation,

Step-2 Equate the demand function and new supply function to get number of units and tax.

Step-3 Calculate the total revue function T that will be product of number of units and tax.

Step-4 Calculate the derivative of revenue function and set equal to zero and solve.

Step-5 Calculate second derivative test to verify the result.

If a quadratic equation is of the form ax2+bx+c=0, then,

x=b±b24ac2a

Calculation:

The provided expression is p=(0.3)q21+10q+500,

After the taxation, the supply function is equal to p=(0.3)q21+10q+500+t. The demand function will meet the new supply function is:

Put the new supply(after taxation) equal to the demand.

(0.3)q21+10q+500+t=500020q0.7q2

Solve to equation further to get the value of t,

Add (500020q0.7q2) on both the sides,

(0.3)q21+10q+500+t(500020q0.7q2)=500020q0.7q2(500020q0.7q2)(0.3)q21+10q+500+t(500020q0.7q2)=0

Deduct t from both the sides,

(0.3)q21+10q+500+t(500020q0

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