Given the demand function:

$q = 44 - \frac{p}{4}$

we need to find:

1. Elasticity of demand $E$
2. Values of $q$ (if any) where total revenue is maximized.

Solution

Step 1: Rewrite the Demand Function

From the demand function, we have: $q = 44 - \frac{p}{4}$

We can rearrange it to express $p$ in terms of $q$: $p = 4(44 - q) = 176 - 4q$

Step 2: Total Revenue Function

Total revenue $R$ is given by $R = p \cdot q$. Substitute $p = 176 - 4q$: $R = (176 - 4q) \cdot q = 176q - 4q^2$

Step 3: Marginal Revenue

To maximize total revenue, we need to find the marginal revenue (the derivative of $R$ with respect to $q$) and set it to zero.

$\frac{dR}{dq} = 176 - 8q$ Setting $\frac{dR}{dq} = 0$: $176 - 8q = 0$ $8q = 176$ $q = 22$

So, total revenue is maximized when $q = 22$.

Step 4: Elasticity of Demand $E$

The elasticity of demand $E$ is defined as: $E = \frac{d q}{d p} \cdot \frac{p}{q}$

From the demand function $q = 44 - \frac{p}{4}$, we find $\frac{dq}{dp} = -\frac{1}{4}$.

Substitute $p = 176 - 4q$ when $q = 22$: $p = 176 - 4 \cdot 22 = 88$

Now calculate $E$ when $q = 22$ and $p = 88$: $E = \left(-\frac{1}{4}\right) \cdot \frac{88}{22} = -1$

Summary of Results

1. Elasticity of demand $E = -1$.
2. Total revenue is maximized when $q = 22$.

Would you like more details on elasticity or maximizing revenue?

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Follow-up Questions

1. How does the elasticity of demand relate to total revenue?
2. Why is demand elasticity important in economics?
3. How would changes in the price affect $q$ in this demand function?
4. What would happen to the total revenue if the demand became more elastic?
5. Could we maximize revenue at other values of $q$ in different demand functions?

Tip

When $E = -1$, demand is unit elastic, meaning revenue does not change with small changes in price.