Let's solve the problem step by step.

The demand function is given as:

$q = 100 - 5p$

Part (a): Find $\varepsilon$ when $p = 15$.

The price elasticity of demand, $\varepsilon$, is given by:

$\varepsilon = \left( \frac{dq}{dp} \right) \cdot \frac{p}{q}$

First, we need to calculate $\frac{dq}{dp}$, the derivative of the demand function with respect to price $p$:

$\frac{dq}{dp} = -5$

Now, substitute $p = 15$ into the demand function to find $q$:

$q = 100 - 5(15) = 100 - 75 = 25$

Now, using the formula for elasticity:

$\varepsilon = (-5) \cdot \frac{15}{25}$

Simplify:

$\varepsilon = -5 \cdot 0.6 = -3$

So, $\varepsilon = -3$.

Part (b): Is the demand elastic, inelastic, or unit elastic?

The rule for elasticity is:

• If $|\varepsilon| > 1$, demand is elastic.
• If $|\varepsilon| = 1$, demand is unit elastic.
• If $|\varepsilon| < 1$, demand is inelastic.

Since $|\varepsilon| = 3$, which is greater than 1, the demand is elastic.

Would you like more details on any part? Here are some related questions you might want to explore:

1. What does it mean when demand is elastic?
2. How would the elasticity change if $p$ were increased or decreased?
3. What factors can affect the elasticity of demand for a commodity?
4. How does elasticity affect pricing decisions for businesses?
5. Can a product ever shift from being elastic to inelastic? If so, how?

Tip: Price elasticity gives insight into how sensitive consumers are to price changes, and can influence decisions on whether to raise or lower prices.