Answer the following questions for the price-demand equation. \( p+0.002 \mathrm{x}=50 \) (A) Express the demand x as a function of the price p . \( \mathrm{x}=\square \) The domain of this function is (Type an inequality or a compound inequality.) (B) Find the elasticity of demand, \( \mathrm{E}(\mathrm{p}) \). \( \mathrm{E}(\mathrm{p})=\square \) (C) What is the elasticity of demand when \( \mathrm{p}=\mathbf{2 5} \) ? The elasticity of demand when \( \mathrm{p}=25 \) is \( \square \) (Type an integer or a simplified fraction.)

To solve for the price-demand equation \( p + 0.002x = 50 \): (A) Rearranging the equation to express demand \( x \) as a function of price \( p \): First, isolate \( x \): \[ 0.002x = 50 - p \] Now, divide by \( 0.002 \): \[ x = \frac{50 - p}{0.002} \] So, the demand function is: \[ \mathrm{x} = 25000 - 500p \] The domain of this function is governed by the values \( p \) can take while keeping \( x \) non-negative: \[ 50 - p \geq 0 \] This implies the inequality: \[ p \leq 50 \] (B) To find the elasticity of demand \( E(p) \), we use the formula: \[ E(p) = \frac{dx/dp \cdot p}{x} \] From the demand function \( x = 25000 - 500p \), we differentiate with respect to \( p \): \[ \frac{dx}{dp} = -500 \] Now we substitute \( x \) and \( \frac{dx}{dp} \): \[ E(p) = \frac{-500 \cdot p}{25000 - 500p} \] (C) Now we calculate the elasticity of demand when \( p = 25 \): Substituting \( p = 25 \) into the \( x \) function: \[ x = 25000 - 500(25) = 25000 - 12500 = 12500 \] Now substituting \( p = 25 \) and \( x = 12500 \) into the elasticity formula: \[ E(25) = \frac{-500 \cdot 25}{12500} \] Calculating \( E(25) \): \[ E(25) = \frac{-12500}{12500} = -1 \] So, the answers are: (A) \( \mathrm{x} = 25000 - 500p \); Domain: \( p \leq 50 \) (B) \( \mathrm{E}(\mathrm{p}) = \frac{-500p}{25000 - 500p} \) (C) The elasticity of demand when \( \mathrm{p} = 25 \) is \( -1 \)