The demand equation provided is:

$(p + 8)q + 4 = 1980$

We are tasked with calculating the elasticity of demand at $p = 82$, classifying the type of elasticity, and determining how a price increase would affect revenue.

(a) Find the Elasticity When $p = 82$

1. First, we need to solve for $q$ in terms of $p$.

$(p + 8)q + 4 = 1980 \quad \text{or} \quad (p + 8)q = 1976$

$q = \frac{1976}{p + 8}$

This is the demand function, $q(p)$. Next, the elasticity of demand, $E(p)$, is given by the formula:

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

First, differentiate $q(p)$ with respect to $p$:

$q(p) = \frac{1976}{p + 8}$

Using the quotient rule for differentiation, we get:

$\frac{dq}{dp} = -\frac{1976}{(p + 8)^2}$

Now plug into the elasticity formula:

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

Substituting $q = \frac{1976}{p + 8}$ and $\frac{dq}{dp} = -\frac{1976}{(p + 8)^2}$:

$E(p) = \left| \frac{p}{\frac{1976}{p + 8}} \cdot \left(-\frac{1976}{(p + 8)^2}\right) \right|$

Simplifying:

$E(p) = \left| \frac{p \cdot (p + 8)^2}{1976} \cdot \left(-\frac{1976}{(p + 8)^2}\right) \right|$

$E(p) = \left| \frac{-p}{p + 8} \right| = \frac{p}{p + 8}$

Now, substitute $p = 82$:

$E(82) = \frac{82}{82 + 8} = \frac{82}{90} \approx 0.91$

Thus, the elasticity at $p = 82$ is approximately $E(82) = 0.91$.

(b) Classifying the Type of Elasticity

The demand is classified based on the elasticity value $E(p)$:

• If $E(p) > 1$, demand is elastic.
• If $E(p) = 1$, demand is unitary.
• If $E(p) < 1$, demand is inelastic.

Since $E(82) = 0.91$, which is less than 1, the demand is inelastic.

(c) Effect of Price Increase on Revenue

For inelastic demand (when $E(p) < 1$), an increase in price increases revenue because the percentage decrease in quantity demanded is smaller than the percentage increase in price.

Thus, an increase in price increases revenue in this case.

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Would you like more details on any part of the calculation?

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Follow-up Questions:

1. What happens to elasticity as $p$ increases?
2. How does elasticity affect total revenue in general?
3. What is the significance of unitary elasticity in pricing decisions?
4. Can demand become elastic at higher prices?
5. How would a change in the demand equation affect elasticity?

Tip: Elasticity helps businesses understand how sensitive customers are to price changes, influencing pricing strategies.