Suppose that the demand D for candy at a theater is inversely related to the price p.
​(a) When the price of candy is ​$3.25 per​ bag, the theater sells 144 bags of candy. Express the demand for candy in terms of its price.​ (b) Determine the number of bags of candy that will be sold if the price is raised to ​$4.00 a bag.

### (a) Expressing the demand for candy in terms of price:
The demand $ D $ is inversely related to the price $ p $. This means that:

\[
D = \frac{k}{p}
\]

where $ k $ is a constant of proportionality. 

Given that when the price $ p = 3.25 $ dollars, the theater sells $ D = 144 $ bags, we can use this information to solve for $ k $:

\[
144 = \frac{k}{3.25}
\]

Solving for $ k $:

\[
k = 144 \times 3.25 = 468
\]

Thus, the demand function in terms of price is:

\[
D(p) = \frac{468}{p}
\]

### (b) Number of bags sold if the price is raised to $ p = 4.00 $:

Using the demand function $ D(p) = \frac{468}{p} $, substitute $ p = 4.00 $:

\[
D(4.00) = \frac{468}{4.00} = 117
\]

So, if the price is raised to $ 4.00 $ dollars per bag, the theater will sell **117 bags of candy**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What happens to the demand if the price is lowered to $2.50 per bag?
2. How would the demand change if the inverse relationship had a different constant?
3. Can this relationship be modeled with a linear function instead of an inverse one? Why or why not?
4. What if the theater introduced a discount, how would that affect the demand?
5. How does this demand function behave as the price approaches zero or infinity?

**Tip:** When dealing with inverse relationships, as one variable increases, the other decreases.

Learn how to express the demand for candy at a theater in terms of its price using inverse proportionality. This example explores the impact of raising the price from $3.25 to $4.00 on the number of bags sold, covering core algebraic concepts. Suitable for Grades 9-10.

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