Calculate the consumer surplus (in dollars) at the unit price = 100 for the demand
equation p = 620e^-3q. (Round your answer to the nearest cent.)

To calculate the **consumer surplus** for a given demand function, we use the following steps:

1. **Demand equation**: $ p = 620 e^{-3q} $
   - $ p $ is the price.
   - $ q $ is the quantity demanded.
   
2. **Consumer surplus formula**:
   \[
   \text{Consumer Surplus} = \int_0^Q \left( p(q) - P \right) dq
   \]
   where:
   - $ p(q) $ is the demand curve,
   - $ P $ is the market price,
   - $ Q $ is the quantity corresponding to the price $ P $.

3. We are given:
   - The market price $ P = 100 $.
   - The demand function $ p(q) = 620 e^{-3q} $.

4. First, find the **quantity $ Q $** corresponding to $ P = 100 $.

   Set $ p(q) = 100 $ and solve for $ q $:
   \[
   100 = 620 e^{-3q}
   \]
   Now let's solve for $ q $.The quantity $ Q $ corresponding to the unit price of $ 100 $ is approximately $ 0.6082 $.

Next, we calculate the **consumer surplus** using the formula:
\[
\text{Consumer Surplus} = \int_0^Q \left( 620 e^{-3q} - 100 \right) dq
\]
This requires computing the integral of the demand function minus the price over the interval from $ 0 $ to $ Q = 0.6082 $. Let's calculate the consumer surplus.The consumer surplus at a unit price of 100 is approximately **$112.52**.

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

Here are some related questions for further exploration:
1. How would the consumer surplus change if the unit price is decreased to 90?
2. What is the total quantity demanded if the price is set to 150?
3. How does a higher decay rate (e.g., 5) in the demand equation affect the consumer surplus?
4. What happens to consumer surplus if the initial price coefficient increases to 700?
5. Can we derive the producer surplus for the same demand function?

**Tip:** Consumer surplus can visually be understood as the area between the demand curve and the price line in a demand-supply graph.

Calculate the consumer surplus (in dollars) at the unit price = 100 for the demand equation p = 620e^-3q. (Round your answer to the nearest cent.)

Learn how to calculate the consumer surplus at a unit price of 100 for the demand function p = 620e^-3q. This problem involves key calculus concepts, including consumer surplus and integration of exponential functions, and is ideal for students in undergraduate economics or calculus.

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