Chapter 2.3, Problem 66E

### Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640

Chapter
Section

### Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

# DEMAND FOR COMMODITIES Assume that the demand function for a certain commodity has the form p = − a x 2 + b (a ≥ 0, b ≥ 0)where x is the quantity demanded, measured in units of a thousand and p is the unit price in dollars. Suppose the quantity demanded is 6000 (x = 6) when the unit price is $8 and 8000 (x = 8) when the unit price is$6. Determine the demand equation. What is the quantity demanded when the unit price is set at \$7.50?

To determine
The demand equation and quantity demanded for the given unit price.

Explanation

Given:

Demand function is,

p=ax2+b

Where,

• x is the quantity demanded.
• p is the unit price in dollars.

Calculation:

Substitute 6 for x and 8 for p in expression p=ax2+b , the expression becomes;

8=a(6)2+b (1)

Substitute 8 for x and 6 for p in expression p=ax2+b , the expression becomes

6=a(8)2+b (2)

Square both sides of equation (1) and (2) and subtract equation (2) from (1) to evaluate the value of a.

6436=(36a+b)(64a+b)28=36a+64a28=28aa=1[Dividebothsidesby28]

Square both sides of equation (1) and substitute 1 for a to evaluate the value of b.

Substitute 1 for a and 100 for b in equation p=ax2+b to find the demand equation

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