For the following demand​ function, find ​(a)​ E, and ​(b) values of q​ (if any) at which total revenue is maximized.
q equals 44 minus StartFraction p Over 4 EndFraction

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.

Find the elasticity of demand (E) and values of q where total revenue is maximized for the demand function q = 44 - p/4. This problem covers important economic concepts such as revenue maximization, elasticity, and marginal revenue. Suitable for high school students in Grades 11-12.

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