Suppose that the total profit in hundreds of dollars from selling x items is given by Upper P left parenthesis x right parenthesis equals 3 x squared minus 4 x plus 9
. 
Complete parts a through d below.
Question content area bottom
Part 1
a. Find the average rate of change of profit as x changes from 2
 
to 4
.
​$1400
 
per item
Part 2
b. Find the average rate of change of profit as x changes from 2
 
to 3
.
​$enter your response here
 
per item

We are given the profit function $ P(x) = 3x^2 - 4x + 9 $, where $ P(x) $ represents the total profit in hundreds of dollars from selling $ x $ items. To find the average rate of change of profit, we will use the formula for the average rate of change over an interval $ [a, b] $:

\[
\text{Average rate of change} = \frac{P(b) - P(a)}{b - a}
\]

### Part 1: $ x $ changes from 2 to 4

We will calculate the average rate of change of profit as $ x $ changes from 2 to 4.

1. Compute $ P(2) $ and $ P(4) $:
   \[
   P(2) = 3(2)^2 - 4(2) + 9 = 3(4) - 8 + 9 = 12 - 8 + 9 = 13
   \]
   \[
   P(4) = 3(4)^2 - 4(4) + 9 = 3(16) - 16 + 9 = 48 - 16 + 9 = 41
   \]

2. Calculate the average rate of change:
   \[
   \frac{P(4) - P(2)}{4 - 2} = \frac{41 - 13}{4 - 2} = \frac{28}{2} = 14
   \]

So, the average rate of change of profit as $ x $ changes from 2 to 4 is **14 hundred dollars per item**, or $1400 per item.

### Part 2: $ x $ changes from 2 to 3

Next, we will calculate the average rate of change of profit as $ x $ changes from 2 to 3.

1. Compute $ P(3) $:
   \[
   P(3) = 3(3)^2 - 4(3) + 9 = 3(9) - 12 + 9 = 27 - 12 + 9 = 24
   \]

2. Calculate the average rate of change:
   \[
   \frac{P(3) - P(2)}{3 - 2} = \frac{24 - 13}{3 - 2} = \frac{11}{1} = 11
   \]

So, the average rate of change of profit as $ x $ changes from 2 to 3 is **11 hundred dollars per item**, or $1100 per item.

---

Would you like further details or have any questions? 

Here are 5 related questions to consider:
1. How does the average rate of change relate to the derivative of the profit function?
2. What is the instantaneous rate of change of profit at $ x = 3 $?
3. How would the average rate of change change if $ x $ were increased to 5?
4. How can you interpret the values of $ P(x) $ economically?
5. What is the significance of the coefficients in the quadratic profit function?

**Tip:** The average rate of change of a function over an interval provides a good estimate of the function's behavior between two points, similar to the slope of a line connecting them.

Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = 3x^2 - 4x + 9. Complete parts a through d below.
Part 1
a. Find the average rate of change of profit as x changes from 2 to 4.
Part 2
b. Find the average rate of change of profit as x changes from 2 to 3.

Learn how to calculate the average rate of change for a quadratic profit function P(x) = 3x^2 - 4x + 9. This problem involves finding the rate of change of profit as x changes over different intervals. Suitable for students in Grades 9-11.

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