- Use the price-demand equation below to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated values of \( p \). \( x=I(p)=510-4 p-0.1 p^{2} \) \( \begin{array}{lll}\text { (A) } p=5 & \text { (B) } p=40 & \text { (C) } p=30\end{array} \)

We begin with the demand function:   x = I(p) = 510 – 4p – 0.1p² The price elasticity of demand is given by:   E(p) = [p / I(p)] × I′(p) Step 1. Find the derivative I′(p).   I′(p) = d/dp [510 – 4p – 0.1p²] = –4 – 0.2p Step 2. Analyze for each value of p. ────────────────────────────── (A) For p = 5: 1. Compute I(5):   I(5) = 510 – 4(5) – 0.1(5²)       = 510 – 20 – 0.1(25)       = 510 – 20 – 2.5       = 487.5 2. Compute I′(5):   I′(5) = –4 – 0.2(5)       = –4 – 1       = –5 3. Elasticity:   E(5) = (5 / 487.5) × (–5)       = –25 / 487.5 ≈ –0.0513 Since the absolute value |E(5)| ≈ 0.0513 is less than 1, the demand is inelastic at p = 5. ────────────────────────────── (B) For p = 40: 1. Compute I(40):   I(40) = 510 – 4(40) – 0.1(40²)       = 510 – 160 – 0.1(1600)       = 510 – 160 – 160       = 190 2. Compute I′(40):   I′(40) = –4 – 0.2(40)       = –4 – 8       = –12 3. Elasticity:   E(40) = (40 / 190) × (–12)       = –480 / 190 ≈ –2.526 Since |E(40)| ≈ 2.53, which is greater than 1, the demand is elastic at p = 40. ────────────────────────────── (C) For p = 30: 1. Compute I(30):   I(30) = 510 – 4(30) – 0.1(30²)       = 510 – 120 – 0.1(900)       = 510 – 120 – 90       = 300 2. Compute I′(30):   I′(30) = –4 – 0.2(30)       = –4 – 6       = –10 3. Elasticity:   E(30) = (30 / 300) × (–10)       = (0.1) × (–10)       = –1 Here, |E(30)| = 1, so the demand exhibits unit elasticity at p = 30. ────────────────────────────── Summary of Answers:  • At p = 5: Inelastic demand (|E| ≈ 0.0513 < 1).  • At p = 40: Elastic demand (|E| ≈ 2.53 > 1).  • At p = 30: Unit elastic demand (|E| = 1).