Homework 5.6

The supply function and demand function for the sale of a certain type of DVD player are given by
S(p)=140140e^(0.005p) and D(p)=490490e^(-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.

$\begin{array}{|rcll|} \hline S(p) &=& D(p) \\ 140140\cdot e^{0.005p} &=& 490490\cdot e^{-0.002p} \quad & | \quad : 140140 \\ e^{0.005p} &=& \frac{490490}{140140} \cdot e^{-0.002p} \\ e^{0.005p} &=& 3.5 \cdot e^{-0.002p} \quad & | \quad \cdot e^{0.002p} \\ e^{0.005p}\cdot e^{0.002p} &=& 3.5 \cdot e^{-0.002p+0.002p} \\ e^{0.005p+0.002p} &=& 3.5\cdot e^0 \quad & | \quad e^0 = 1 \\ e^{0.007p} &=& 3.5 \quad & | \quad \ln() \text{ both sides } \\ \ln(~e^{0.007p}~) &=& \ln(3.5) \\ 0.007 p\cdot \ln(e) &=& \ln(3.5) \quad & | \quad \ln(e) = 1\\ 0.007\cdot p &=& \ln(3.5) \quad & | \quad : 0.007 \\ p &=& \frac{ \ln(3.5) }{ 0.007 } \\ p &=& \frac{ 1.25276296850 }{ 0.007 } \\ \mathbf{p} &\mathbf{=} & \mathbf{178.966138356} \\ \hline \end{array}$

The equilibrium price is 178.97