The daily sales of t-shirts can be represented by the linear demand equation \( q=590-4 p \). Find the price elasticity of demand when the price is \( \$ 45 \) per \( t \)-shirt. (Round your answer to 2 decimal places.)

To calculate the price elasticity of demand \( E \) when the price is \( \$45 \) per t-shirt, we'll use the following formula: \[ E = \frac{dq}{dp} \times \frac{p}{q} \] **Step 1: Determine \( \frac{dq}{dp} \)** Given the demand equation: \[ q = 590 - 4p \] The derivative of \( q \) with respect to \( p \) is: \[ \frac{dq}{dp} = -4 \] **Step 2: Calculate \( q \) at \( p = \$45 \)** \[ q = 590 - 4 \times 45 = 590 - 180 = 410 \] **Step 3: Plug values into the elasticity formula** \[ E = (-4) \times \frac{45}{410} = -\frac{180}{410} \approx -0.439 \] **Step 4: Round to two decimal places** \[ E \approx -0.44 \] **Interpretation:** The price elasticity of demand is \(-0.44\). This indicates that the demand is inelastic at this price point, meaning that a 1% increase in price would lead to approximately a 0.44% decrease in quantity demanded. **Final Answer:** \(-0.44\)