Questions: The demand for a product is given by x=F(p)=1840/(5+ln (2+p)). If p=30, determine E and interpret the results.

Solution Steps

To solve this problem, we need to calculate the elasticity of demand $E$ at a given price $p = 30$. The elasticity of demand is given by the formula $E = \frac{p}{x} \cdot \frac{dx}{dp}$. First, we need to find $x$ by substituting $p = 30$ into the demand function. Then, we calculate the derivative $\frac{dx}{dp}$ of the demand function. Finally, we substitute these values into the elasticity formula to find $E$.

We start by substituting $p = 30$ into the demand function $x = \frac{1840}{5 + \ln(2 + p)}$:

$$x = \frac{1840}{5 + \ln(32)} \approx \frac{1840}{5 + 3.465736} \approx \frac{1840}{8.465736} \approx 217.4$$

Next, we find the derivative of the demand function with respect to $p$:

$$\frac{dx}{dp} = -\frac{1840}{(p + 2)(5 + \ln(2 + p))^2}$$

Substituting $p = 30$:

$$\frac{dx}{dp} = -\frac{1840}{(32)(5 + \ln(32))^2} \approx -\frac{1840}{32 \cdot (8.465736)^2} \approx -\frac{1840}{32 \cdot 71.703} \approx -\frac{1840}{2294.464} \approx -0.802$$

Now we can calculate the elasticity of demand $E$ using the formula:

$$E = \frac{p}{x} \cdot \frac{dx}{dp}$$

Substituting the values we found:

$$E = \frac{30}{217.4} \cdot (-0.802) \approx -0.1107$$

Final Answer