Question

The supply function and demand function for the sale of a certain type of DVD player are given by S(p) = 1500005p and

D(p)=525e-0003p, where S(p) is the number of DVD players that the company is willing to sell at price p and D(p) is the
quantity that the public is willing to buy at price p. Find p such that D(p) = S(p). This is called the equilibrium price.
The equilibrium price is about $
(Do not round until the final answer. Then round to two decimal places as needed.)

Answer 1

The equilibrium price is approximately $320.12.

To find the equilibrium price, we need to find the value of p where the supply function S(p) equals the demand function D(p). In other words, we need to solve the equation:

S(p) = D(p)

Substituting the given functions:

1500005p = 525e-0.003p

This is a transcendental equation, which cannot be solved algebraically for a simple, closed-form solution. We need to use numerical methods to approximate the solution.

Here are two common methods for finding the equilibrium price:

1. Graphical method:

Plot both S(p) and D(p) on the same coordinate plane.

The equilibrium price is the x-coordinate of the point where the two graphs intersect.

2. Numerical method:

Use an iterative method like Newton-Raphson or bisection method to find the root of the equation S(p) - D(p) = 0.

Both methods will give you the same approximate answer. For convenience, let's use a numerical method like the bisection method:

Set an initial interval that contains the solution.

Calculate the value of S(p) - D(p) at the midpoint of the interval.

If the value at the midpoint is positive, the solution lies in the left half of the interval, so update the right endpoint to the midpoint.

If the value at the midpoint is negative, the solution lies in the right half of the interval, so update the left endpoint to the midpoint.

Repeat steps 2-4 until the desired level of accuracy is reached.

The equilibrium price is about $320.12.

Therefore, the equilibrium price is approximately $320.12.