The supply function and demand function for the sale of a certain type of DVD player are given by S(p)=130e0.004p and D(p)=416e−0.002p, 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.

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ajdonis ajdonis · Answer 1 · 2020-12-08T00:30:22+01:00

Answer:

p=$193.86 (assuming we are talking about dollars)

Step-by-step explanation:

In order to solve this problem, we must set each function equal to each other, so we end up with the following equation we must solve for p:

[tex]130e^{0.004p}=416e^{-0.002p}[/tex]

So next, we divide both sides of the equation by 130 and by [tex]e^{-0.002p}[/tex]

So we get:

[tex]\frac{e^{0.004p}}{e^{-0.002p}}=\frac{416}{130}[/tex]

and we simplify, so we get:

[tex]e^{0.004p}e^{0.002p}=3.2[/tex]

which can be further simplified to:

[tex]e^{0.006p}=3.2[/tex]

and next, we take the natural logarithm to both sides, so we get:

0.006p=ln(3.2)

and finally we divide both sides of the equation by 0.006 so we get:

[tex]p=\frac{ln(3.2)}{0.006}[/tex]

and simplify so we get our answer:

p=$193.86 (in the case that p is given in dollars)