The demand for a certain fraternity's plastic brownie dishes is
q(p) = 363,609 − (p + 1)2
where q represents the number of brownie dishes that the fraternity can sell each month at a price of p¢. Use this function to determine the following.
(a)
Determine the number of brownie dishes the fraternity can sell each month if the price is set at 80¢.
brownie dishes
(b)
Determine the number of brownie dishes they can unload each month if they give them away.
brownie dishes
(c)
Determine the lowest price at which the fraternity will be unable to sell any dishes.
¢

The given demand function is $ q(p) = 363,609 - (p + 1)^2 $, where $ p $ represents the price in cents, and $ q(p) $ represents the number of brownie dishes the fraternity can sell at that price.

### (a) Determine the number of brownie dishes the fraternity can sell if the price is set at 80¢.
To solve this, we substitute $ p = 80 $ into the demand function:

\[
q(80) = 363,609 - (80 + 1)^2
\]
\[
q(80) = 363,609 - 81^2
\]
\[
q(80) = 363,609 - 6,561
\]
\[
q(80) = 357,048
\]

Thus, the fraternity can sell **357,048 brownie dishes** at 80¢ each.

### (b) Determine the number of brownie dishes they can unload each month if they give them away (price = 0¢).
For this, we substitute $ p = 0 $ into the demand function:

\[
q(0) = 363,609 - (0 + 1)^2
\]
\[
q(0) = 363,609 - 1^2
\]
\[
q(0) = 363,609 - 1
\]
\[
q(0) = 363,608
\]

Thus, the fraternity can sell **363,608 brownie dishes** if they give them away.

### (c) Determine the lowest price at which the fraternity will be unable to sell any dishes.
The fraternity will be unable to sell any dishes when $ q(p) = 0 $. We set the demand function equal to 0 and solve for $ p $:

\[
0 = 363,609 - (p + 1)^2
\]
\[
(p + 1)^2 = 363,609
\]
\[
p + 1 = \sqrt{363,609}
\]
\[
p + 1 = 603
\]
\[
p = 602
\]

Thus, the fraternity will be unable to sell any dishes if the price is **602¢** (or $6.02).

---

Do you need further details or have any questions about this problem?

Here are 5 related questions to consider:
1. How does the demand change if the price is increased to 100¢?
2. What is the price at which only 100,000 dishes will be sold?
3. Can you find the price that maximizes the number of dishes sold?
4. How many dishes can be sold if the price is set at 500¢?
5. What happens to the demand if the price becomes negative?

**Tip:** Always verify units (like cents vs dollars) when plugging into formulas to avoid miscalculations.

The demand for a certain fraternity's plastic brownie dishes is q(p) = 363,609 − (p + 1)^2 where q represents the number of brownie dishes that the fraternity can sell each month at a price of p¢. Use this function to determine the following. (a) Determine the number of brownie dishes the fraternity can sell each month if the price is set at 80¢. (b) Determine the number of brownie dishes they can unload each month if they give them away. (c) Determine the lowest price at which the fraternity will be unable to sell any dishes.

Explore how to use the quadratic demand function q(p) = 363,609 − (p + 1)^2 to calculate the number of brownie dishes a fraternity can sell at different prices. Learn how to solve for the number of dishes sold at 80¢, at 0¢ (when given away), and find the price where sales drop to zero.

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