Question 1179800: A kids' organization sold 730 bars last year at $ 2.25 per candy bar. Research has suggested that for each $ 0.45 price increase, they will sell 40 less candy bars. They want to know what price should they sell their candy bars at to maximize their revenue.
Please help them out, showing your work neatly in the following steps. And please label the steps clearly with the letters A through F
A) Define your variables. This means that for every letter you use to represent a value, you communicate what it equals in words.
B) Find the linear equation that relates the price and the number sold.
C) Determine a function that represents the Revenue.
(Remember, Revenue = (Price)*(Number Sold)).
D) Graph your revenue function on your calculator or Desmos or something similar and on your paper, labeling the axes and the maximum point.
E) Using your calculator or Desmos or something similar, use your graph to determine the price that maximizes the revenue for the organization.
Do NOT use calculus to solve this problem. (In other words, if you seek out help, don't let those that help you show you how to do it with calculus. If they use the word "derivative," that's calculus.)
The price of a candy bar that maximizes the revenue is $ ______ Round to the nearest cent.
F) Also, determine the number they will sell at this price:
The number of candy bars they will sell at that price is _____ Round to the nearest whole number.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.**
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