The dd for a commodity is given by 𝑥 = 48 − 3𝑝 . Find the point elasticity
2
of demand when p=3. If this price of 3 is decreased by 4%, determine the
relative increase in dd and hence an approximation to the elasticity of
demand.

### 1. Point Elasticity of Demand

The demand function is given by:

\[
x = 48 - 3p^2
\]

To find the point elasticity of demand at \( p = 3 \), we use the formula for point elasticity of demand:

\[
E(p) = \frac{d\ln(x)}{d\ln(p)} = \frac{p}{x} \cdot \frac{dx}{dp}
\]

#### Step 1: Compute \( \frac{dx}{dp} \)

First, differentiate the demand function \( x = 48 - 3p^2 \) with respect to \( p \):

\[
\frac{dx}{dp} = -6p
\]

#### Step 2: Evaluate \( \frac{dx}{dp} \) at \( p = 3 \)

Substituting \( p = 3 \):

\[
\frac{dx}{dp} = -6 \times 3 = -18
\]

#### Step 3: Compute \( x \) at \( p = 3 \)

Substitute \( p = 3 \) into the demand function:

\[
x = 48 - 3(3^2) = 48 - 27 = 21
\]

#### Step 4: Calculate the Point Elasticity

Now, using the point elasticity formula:

\[
E(3) = \frac{3}{21} \cdot (-18) = \frac{3 \times (-18)}{21} = -\frac{54}{21} = -\frac{18}{7} \approx -2.57
\]

So, the point elasticity of demand at \( p = 3 \) is approximately \( -2.57 \).

### 2. Relative Increase in Demand Due to a 4% Decrease in Price

#### Step 1: Calculate the new price after a 4% decrease

The price is decreased by 4%:

\[
\text{New price} = 3 \times (1 - 0.04) = 3 \times 0.96 = 2.88
\]

#### Step 2: Compute the new demand at \( p = 2.88 \)

Substitute \( p = 2.88 \) into the demand function:

\[
x_{new} = 48 - 3(2.88^2) = 48 - 3(8.2944) = 48 - 24.8832 = 23.1168
\]

#### Step 3: Calculate the relative increase in demand

The original demand at \( p = 3 \) was \( x = 21 \). The relative increase in demand is:

\[
\text{Relative increase} = \frac{x_{new} - x_{old}}{x_{old}} \times 100 = \frac{23.1168 - 21}{21} \times 100 \approx \frac{2.1168}{21} \times 100 \approx 10.08\%
\]

So, the relative increase in demand is approximately 10.08%.

### 3. Approximation to the Elasticity of Demand

Using the formula for elasticity approximation:

\[
E \approx \frac{\text{Percentage change in demand}}{\text{Percentage change in price}} = \frac{10.08\%}{4\%} \approx 2.52
\]

So, the approximate elasticity of demand is \( -2.52 \).

### Summary
- **Point Elasticity at \( p = 3 \)**: \( -2.57 \)
- **Relative increase in demand**: \( 10.08\% \)
- **Approximate Elasticity**: \( -2.52 \)

Would you like more details or have any questions?

#### 5 Related Questions:
1. How does elasticity of demand change with different price levels?
2. What are the factors that influence the elasticity of a commodity?
3. How can businesses use elasticity of demand in pricing strategies?
4. What are the implications of a negative elasticity value?
5. How does elasticity affect total revenue?

#### Tip:
Elasticity of demand helps predict how changes in price affect the quantity demanded, crucial for setting prices strategically.

The dd for a commodity is given by 𝑥 = 48 − 3𝑝^2. Find the point elasticity of demand when p=3. If this price of 3 is decreased by 4%, determine the relative increase in dd and hence an approximation to the elasticity of demand.

Learn how to calculate the point elasticity of demand for the function x = 48 − 3p² at p = 3, and determine the relative increase in demand after a 4% price decrease. This problem involves differentiation and elasticity concepts, suitable for advanced high school or undergraduate students.

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