Questions: If a price-demand equation is solved for p, then price is expressed as p=g(x) and x becomes the independent variable. In this case, it can be shown that the elasticity of demand is given by E(x)=-g(x)/(x g'(x)). Use the given price-demand equation to find the values of x for which demand is elastic and for which demand is inelastic. p=g(x)=9000-0.1 x^2 The values of x for which the demand is elastic are (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Solution Steps

To determine the values of \( x \) for which the demand is elastic or inelastic, we need to calculate the elasticity of demand \( E(x) \) using the given price-demand equation \( p = g(x) = 9000 - 0.1x^2 \). First, find the derivative \( g'(x) \). Then, substitute \( g(x) \) and \( g'(x) \) into the elasticity formula \( E(x) = -\frac{g(x)}{x g'(x)} \). The demand is elastic when \( E(x) > 1 \) and inelastic when \( E(x) < 1 \). Solve the inequality to find the intervals for \( x \).

The price-demand equation is given by: \[ p = g(x) = 9000 - 0.1x^2 \]

The derivative of \( g(x) \) with respect to \( x \) is: \[ g'(x) = -0.2x \]

The elasticity of demand \( E(x) \) is defined as: \[ E(x) = -\frac{g(x)}{x g'(x)} = -\frac{9000 - 0.1x^2}{x \cdot (-0.2x)} = \frac{5(9000 - 0.1x^2)}{x^2} \]

To find the values of \( x \) for which demand is elastic, we solve the inequality: \[ E(x) > 1 \] This leads to: \[ x < 173.2051 \]

To find the values of \( x \) for which demand is inelastic, we solve the inequality: \[ E(x) < 1 \] This leads to: \[ 173.2051 < x \]

Final Answer