Questions: For the demand function q=D(x)=300/x, find the following. a) The elasticity b) The elasticity at x=7, stating whether the demand is elastic, inelastic, or has unit elasticity c) The value(s) of x for which total revenue is a maximum (assume that x is in dollars) a) Find the equation for elasticity. b) Find the elasticity at the given price, stating whether the demand is elastic, inelastic, or has unit elasticity. Is the demand at x=7 elastic, inelastic, or does it have unit elasticity? A. elastic B. unit elasticity C. inelastic

Solution Steps

To solve the given problem, we need to follow these steps:

a) The elasticity of demand, \( E(x) \), is calculated using the formula: \[ E(x) = -\frac{x}{q} \cdot \frac{dq}{dx} \] where \( q = D(x) = \frac{300}{x} \). First, find the derivative \( \frac{dq}{dx} \).

b) Substitute \( x = 7 \) into the elasticity formula to find \( E(7) \). Determine if the demand is elastic, inelastic, or has unit elasticity based on the value of \( E(7) \).

c) To find the value of \( x \) for which total revenue is maximized, use the relationship between elasticity and revenue. Total revenue is maximized when elasticity \( E(x) = 1 \).

The elasticity of demand is given by the formula: \[ E(x) = -\frac{x}{q} \cdot \frac{dq}{dx} \] For the demand function \( q = D(x) = \frac{300}{x} \), we find the derivative: \[ \frac{dq}{dx} = -\frac{300}{x^2} \] Substituting \( q \) and \( \frac{dq}{dx} \) into the elasticity formula, we have: \[ E(x) = -\frac{x}{\frac{300}{x}} \cdot \left(-\frac{300}{x^2}\right) = 1 \]

Substituting \( x = 7 \) into the elasticity formula: \[ E(7) = 1 \] Since \( E(7) = 1 \), the demand at \( x = 7 \) has unit elasticity.

Total revenue is maximized when the elasticity \( E(x) = 1 \). From our calculations, we found that: \[ E(x) = 1 \] This indicates that there are no specific values of \( x \) that yield a maximum total revenue under the given demand function.

Final Answer