Question 1179800
Absolutely! Let's help the kids' organization maximize their revenue.

**A) Define Your Variables:**

* **p:** The price of a candy bar in dollars.
* **n:** The number of candy bars sold.
* **x:** The number of $0.45 price increases.
* **R:** The total revenue in dollars.

**B) Find the Linear Equation that Relates the Price and the Number Sold:**

* **Price:** The initial price is $2.25, and it increases by $0.45 for each price increase (x):
    * p = 2.25 + 0.45x
* **Number Sold:** The initial number sold is 730, and it decreases by 40 for each price increase (x):
    * n = 730 - 40x

**C) Determine a Function that Represents the Revenue:**

* Revenue = (Price) * (Number Sold)
* R = p * n
* R = (2.25 + 0.45x) * (730 - 40x)
* R = 1642.5 - 90x + 328.5x - 18x²
* R = -18x² + 238.5x + 1642.5

**D) Graph Your Revenue Function:**

* You can use a graphing calculator or Desmos to graph the function R = -18x² + 238.5x + 1642.5.
* **On Paper:**
    * Label the x-axis as "Number of Price Increases (x)"
    * Label the y-axis as "Revenue (R)"
    * The graph will be a parabola opening downwards.
    * The maximum point (vertex) will be the point where the revenue is maximized.

**E) Using Your Graph to Determine the Price that Maximizes the Revenue:**

* **Find the x-coordinate of the vertex:**
    * Use the "maximum" or "vertex" function on your calculator or Desmos.
    * The x-coordinate of the vertex represents the number of price increases that maximize revenue.
    * x ≈ 6.625
* **Calculate the price (p):**
    * p = 2.25 + 0.45x
    * p = 2.25 + 0.45(6.625)
    * p = 2.25 + 2.98125
    * p ≈ 5.23125
* **Round to the nearest cent:**
    * p ≈ $5.23

**The price of a candy bar that maximizes the revenue is $5.23.**

**F) Also, Determine the Number They Will Sell at This Price:**

* **Calculate the number of candy bars (n):**
    * n = 730 - 40x
    * n = 730 - 40(6.625)
    * n = 730 - 265
    * n = 465

**The number of candy bars they will sell at that price is 465.**