Question

The supply function and demand function for the sale of a certain type of DVD player are given by S(p) = 150e0.005p and

D(p) = 525e-0.003p
P, 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

Setting demand equal to supply, solving for p gives an equilibrium price of approximately $95.56.

To find the equilibrium price, set the demand function equal to the supply function:

D(p) = S(p)

525e^(-0.003p) = 150e^(0.005p)

Now, divide both sides by 150:

(525/150)e^(-0.003p) = e^(0.005p)

Simplify the left side:

3.5e^(-0.003p) = e^(0.005p)

Take the natural logarithm (ln) of both sides:

ln(3.5e^(-0.003p)) = ln(e^(0.005p))

Use logarithm properties to simplify:

ln(3.5) - 0.003p = 0.005p

Isolate p:

-0.003p - 0.005p = ln(3.5)

Combine like terms:

-0.008p = ln(3.5)

Finally, solve for p:

p = ln(3.5)/(-0.008)

Use a calculator to find the numerical value. The equilibrium price is approximately p ≈ 95.56.

So, the equilibrium price is about $95.56 when rounded to two decimal places.