Calculate the consumers' surplus (in dollars) at the indicated unit price
p
for the demand equation.
p = 14 − 2q1⁄3;  p = 8

To find the consumer's surplus at the indicated unit price $ p = 8 $ for the demand equation $ p = 14 - 2q^{1/3} $, we will follow these steps:

1. **Solve for the quantity $ q $ when $ p = 8 $**:
   
   The demand equation is:
   \[
   p = 14 - 2q^{1/3}
   \]
   Substitute $ p = 8 $:
   \[
   8 = 14 - 2q^{1/3}
   \]
   Solve for $ q^{1/3} $:
   \[
   2q^{1/3} = 14 - 8
   \]
   \[
   2q^{1/3} = 6
   \]
   \[
   q^{1/3} = 3
   \]
   \[
   q = 3^3
   \]
   \[
   q = 27
   \]

2. **Find the consumer's surplus**:
   
   The consumer's surplus is the area between the demand curve and the price level up to the quantity demanded. It can be calculated using the integral of the demand function from 0 to $ q $ minus the total revenue (price times quantity).

   The demand function can be rewritten as:
   \[
   q = \left( \frac{14 - p}{2} \right)^3
   \]
   The consumer's surplus (CS) is given by:
   \[
   \text{CS} = \int_0^{27} (14 - 2q^{1/3}) \, dq - (8 \times 27)
   \]

   First, calculate the integral:
   \[
   \int_0^{27} (14 - 2q^{1/3}) \, dq = \int_0^{27} 14 \, dq - \int_0^{27} 2q^{1/3} \, dq
   \]
   Evaluate the first part:
   \[
   \int_0^{27} 14 \, dq = 14q \bigg|_0^{27} = 14 \times 27 = 378
   \]
   Evaluate the second part:
   \[
   \int_0^{27} 2q^{1/3} \, dq = 2 \int_0^{27} q^{1/3} \, dq
   \]
   Using the power rule for integration:
   \[
   \int q^{1/3} \, dq = \frac{3}{4} q^{4/3} \bigg|_0^{27}
   \]
   So:
   \[
   2 \int_0^{27} q^{1/3} \, dq = 2 \cdot \frac{3}{4} q^{4/3} \bigg|_0^{27} = \frac{3}{2} q^{4/3} \bigg|_0^{27}
   \]
   \[
   \frac{3}{2} \times 27^{4/3} = \frac{3}{2} \times (27^{1/3})^4 = \frac{3}{2} \times 3^4 = \frac{3}{2} \times 81 = 121.5
   \]

   Therefore, the integral part is:
   \[
   \int_0^{27} (14 - 2q^{1/3}) \, dq = 378 - 121.5 = 256.5
   \]

   Now, subtract the total revenue:
   \[
   \text{Total revenue} = 8 \times 27 = 216
   \]
   
   Finally, the consumer's surplus is:
   \[
   \text{CS} = 256.5 - 216 = 40.5
   \]

So, the consumer's surplus at $ p = 8 $ is $ \boxed{40.5} $ dollars.

Would you like more details or have any questions about this solution?

Here are some related questions:
1. How would the consumer's surplus change if the unit price was 10 instead of 8?
2. What is the total revenue when the unit price is 8?
3. How does the consumer's surplus change as the unit price approaches 0?
4. How do we interpret the consumer's surplus in terms of economic welfare?
5. How would the consumer's surplus change if the demand equation was $ p = 14 - 3q^{1/3} $?
6. What are the steps to derive the integral for consumer's surplus?
7. How does the demand curve reflect consumer preferences?
8. What are other methods to calculate consumer's surplus?

**Tip:** When integrating functions with fractional powers, converting them into an exponent form and using the power rule can simplify the calculations.

Learn how to calculate consumer's surplus at a unit price of $8 using the demand equation p = 14 - 2q^(1/3). Follow step-by-step instructions with detailed integral calculus and economic analysis. Suitable for college-level students studying demand and supply analysis.

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