Problem
Find the consumers' surplus and the producers' surplus at the equilibrium price level for the given price-demand and price-supply equations. Include a graph that identifies the consumers' surplus and the producers' surplus. Round all values to the nearest integer. \[ p=D(x)=140 e^{-0.005 x} ; p=S(x)=30 e^{0.005 x} \] The value of $\mathrm{x}$ at equilibrium is $\square$.
Solution
Step 1 ：Set the demand equation equal to the supply equation: \(140 e^{-0.005 x} = 30 e^{0.005 x}\)
Step 2 ：Solve the equation for x to find the equilibrium quantity.
Step 3 ：The equilibrium quantity, x, is approximately 154.044504094715.
Step 4 ：Round the equilibrium quantity to the nearest integer: \(\boxed{154}\)
