   Chapter 11.5, Problem 19E ### 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 weekly demand function is p   =   200   − 2 q 2 and the supply function before taxation is p =   20   +   3 q , what tax per item will maximize the total tax revenue?

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=2002q2 and the supply function before taxation is p=20+3q.

Explanation

Given Information:

The provided expression is the demand function for a fixed period of time is given by p=2002q2 and the supply function before taxation is p=20+3q.

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.

Calculation:

The provided expression is p=20+3q,

After the taxation, the supply function is equal to p=20+3q+t. The demand function will meet the new supply function is:

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

20+3q+t=2002q2

Solve to equation further to get the value of t,

Add 200+2q2 on both the sides,

20+3q+t+(200+2q2)=2002q2+(200+2q2)20+3q+t+(200+2q2)=0

Deduct t from both the sides,

20+3q+t+(200+2q2)t=0t

Multiply by -1 on both the sides,

1803q2q2=t

Then, compute the total tax by the formula T=tq,

T=q(1803q2q2)=(180q3q22q3)

Tax is maximized as follows:

The first derivative of T=(180q3q22q3) is as follows:

T(q)=ddq(180<

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