A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. The company uses the following quadratic function to model the revenue  
R
 , in millions of dollars, as a function of the price per item  
p
 , in dollars:
R
(
p
)
=
−
p
2
+
5
p
,
 
defined for  
0
≤
p
≤
5
 . Predict the revenue if the company sells the item at a price of  
p
=
$
1
 .

Do not include the dollar sign in your answer. Submit your answer as a number that represents the revenue in millions of dollars. For instance, if your answer is  
9
 , then report your answer as such. Do not convert it to a dollar amount such as  
$
9
,
000
,
000
 .

The revenue function given by the company is:

\[
R(p) = -p^2 + 5p
\]

We need to predict the revenue when the price per item $ p = 1 $.

Substituting $ p = 1 $ into the function:

\[
R(1) = - (1)^2 + 5 \times 1
\]

\[
R(1) = -1 + 5
\]

\[
R(1) = 4
\]

So, the predicted revenue when the company sells the item at a price of $1 is:

\[
\text{Revenue} = 4 \text{ million dollars}
\]

Would you like more details, or do you have any questions?

Here are some related questions to explore further:
1. What is the maximum revenue the company can achieve, and at what price?
2. How does the revenue change if the price increases to $3?
3. What is the price at which the revenue becomes zero?
4. How does the revenue function behave as the price approaches its upper limit of $5?
5. What is the significance of the coefficients in the quadratic function?

**Tip:** For quadratic functions modeling revenue or profit, the maximum or minimum value often occurs at the vertex, which can be calculated using $-\frac{b}{2a}$.

A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. The company uses the following quadratic function to model the revenue R, in millions of dollars, as a function of the price per item p, in dollars: R(p) = -p^2 + 5p, defined for 0 ≤ p ≤ 5. Predict the revenue if the company sells the item at a price of p = $1.

Learn how to predict a company's revenue using a quadratic function. This problem models revenue as R(p) = -p^2 + 5p and asks you to find the revenue when the price is set at $1. Explore the mathematical concepts of quadratic functions and revenue optimization. Suitable for Grades 8-10.

Math Problem Statement

A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. The company uses the following quadratic function to model the revenue
R , in millions of dollars, as a function of the price per item
p , in dollars: R ( p )

defined for
0 ≤ p ≤ 5 . Predict the revenue if the company sells the item at a price of
p

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