Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = 5x7 - 5× + 5. Complete parts a through d belo
a. Find the average rate of change of profit as x changes from 3 to 5. per item

The average rate of change of a function \( P(x) \) over an interval \([a, b]\) is calculated by the formula:

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

In this case, the profit function is given as \( P(x) = 5x^2 - 5x + 5 \), and we are asked to find the average rate of change as \( x \) changes from 3 to 5. Here, \( a = 3 \) and \( b = 5 \).

### Step 1: Find \( P(5) \)

\[
P(5) = 5(5)^2 - 5(5) + 5
\]
\[
= 5(25) - 25 + 5 = 125 - 25 + 5 = 105
\]

### Step 2: Find \( P(3) \)

\[
P(3) = 5(3)^2 - 5(3) + 5
\]
\[
= 5(9) - 15 + 5 = 45 - 15 + 5 = 35
\]

### Step 3: Calculate the average rate of change

\[
\text{Average rate of change} = \frac{P(5) - P(3)}{5 - 3} = \frac{105 - 35}{5 - 3} = \frac{70}{2} = 35
\]

Thus, the average rate of change of profit as \( x \) changes from 3 to 5 is 35 (hundreds of dollars per item).

Would you like more details or have any questions?

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### 5 Related Questions:
1. What is the interpretation of the average rate of change in this context?
2. How would the average rate of change differ if \( x \) changed from 5 to 7?
3. How do you compute the instantaneous rate of change for \( P(x) \)?
4. What happens to the profit when \( x = 0 \)? Is there a minimum number of items that should be sold for profit?
5. Can we use this average rate of change to predict profit for future values of \( x \)?

### Tip:
The average rate of change can help you understand how quickly profit is growing or shrinking over a range of values. For exact growth at any point, though, you need the derivative.

Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = 5x^2 - 5x + 5. Find the average rate of change of profit as x changes from 3 to 5.

Learn how to find the average rate of change of a quadratic profit function P(x) = 5x^2 - 5x + 5 as x changes from 3 to 5. This problem involves applying the average rate of change formula and interpreting it in a real-world profit context. Suitable for high school students in Grades 10-12.

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