Questions: For the demand function q=D(p)=100/(p+3)^5, find the following. a) The elasticity b) The elasticity at p=8, stating whether the demand is elastic, inelastic or has unit elasticity c) The value(s) of p for which total revenue is a maximum (assume that p is in dollars) a) Find the equation for elasticity.

For the demand function q=D(p)=100/(p+3)^5, find the following. a) The elasticity b) The elasticity at p=8, stating whether the demand is elastic, inelastic or has unit elasticity c) The value(s) of p for which total revenue is a maximum (assume that p is in dollars) a) Find the equation for elasticity.
Transcript text: For the demand function $\mathrm{q}=\mathrm{D}(\mathrm{p})=\frac{100}{(\mathrm{p}+3)^{5}}$, find the following. a) The elasticity b) The elasticity at $p=8$, stating whether the demand is elastic, inelastic or has unit elasticity c) The value(s) of $p$ for which total revenue is a maximum (assume that $p$ is in dollars) a) Find the equation for elasticity. \[ E(p)=\square \]
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Solution

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Solution Steps

Step 1: Finding the Elasticity of Demand, \(E(p)\)

The elasticity of demand is given by \(E(p) = \frac{p}{q} \times \frac{dq}{dp} = - \frac{5 p}{p + 3}\).

Step 2: Elasticity at a Specific Price, \(p_0\)

At \(p_0 = 8\), the elasticity of demand is \(E(p_0) = -3.64\), which classifies the demand as inelastic.

Step 3: Maximizing Total Revenue

The price that maximizes total revenue is found by solving \(\frac{d(TR)}{dp} = 0\). The critical point is \(p = 0.75\).

Final Answer:

The elasticity of demand at \(p_0 = 8\) is -3.64, classified as inelastic. The price that maximizes total revenue is \(p = 0.75\).

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