The demand function for a product is modeled by
p = 13,000

1 − 
7
7 + e−0.001x

.
Find the price p (in dollars) of the product when the quantity demanded is x = 1000 units and x = 1500 units. What is the limit of the price as x increases without bound?
(a)    x = 1000 units (Round your answer to two decimal places.)
$  



(b)    x = 1500 units (Round your answer to two decimal places.)
$  



(c) What is the limit of the price as x increases without bound?
$  


The demand function is given as:

\[
p(x) = \frac{13,000}{1 - \frac{7}{7 + e^{-0.001x}}}
\]

Let's solve for the price $ p $ when $ x = 1000 $, $ x = 1500 $, and determine the limit as $ x \to \infty $.

### (a) For $ x = 1000 $:

We substitute $ x = 1000 $ into the formula and calculate the corresponding price.

\[
p(1000) = \frac{13,000}{1 - \frac{7}{7 + e^{-0.001 \cdot 1000}}}
\]

### (b) For $ x = 1500 $:

Substitute $ x = 1500 $ into the formula.

\[
p(1500) = \frac{13,000}{1 - \frac{7}{7 + e^{-0.001 \cdot 1500}}}
\]

### (c) Limit of the price as $ x \to \infty $:

As $ x \to \infty $, $ e^{-0.001x} \to 0 $. Thus, the demand function simplifies to:

\[
p(x) \to \frac{13,000}{1 - \frac{7}{7}} = \frac{13,000}{1 - 1} = \infty
\]

So, the price increases without bound as $ x \to \infty $.

Now, I will calculate the values for parts (a) and (b).The calculated prices are:

(a) For $ x = 1000 $ units, the price is approximately $ 260,363.65 $ dollars.

(b) For $ x = 1500 $ units, the price is approximately $ 420,833.71 $ dollars.

(c) As $ x \to \infty $, the price $ p(x) \to \infty $ as well, meaning the price increases without bound.

Let me know if you would like more details or have any questions. 

Here are some related questions:
1. What happens to the price if the rate constant in the exponential changes?
2. How does the price behave for small values of $x$?
3. Can you explain how the exponential term affects the price growth?
4. What if the coefficient in front of $ e^{-0.001x} $ were different, how would it impact $ p(x) $?
5. How would the demand function change if a different constant was used instead of 13,000?

**Tip:** Always verify exponential terms in demand functions carefully—they control the rapidity of price changes with demand.

The demand function for a product is modeled by p = 13,000 / (1 - 7 / (7 + e^(-0.001x))). Find the price p (in dollars) of the product when the quantity demanded is x = 1000 units and x = 1500 units. What is the limit of the price as x increases without bound? (a) x = 1000 units (Round your answer to two decimal places.) (b) x = 1500 units (Round your answer to two decimal places.) (c) What is the limit of the price as x increases without bound?

Learn how to calculate the price of a product using the demand function p = 13,000 / (1 - 7 / (7 + e^(-0.001x))) at quantities of 1000 and 1500 units. This problem covers key concepts such as exponential functions, limits, and demand functions. Suitable for students in Grades 11-12 and early college-level economics and calculus courses.

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