Questions: Evaluating and Solving Quadratic Functions It is very important in business for companies to determine their profit in different situations. Profit is revenue (money earned) minus costs (money spent). A company's revenue earned from selling x items is given by the function R(x)=490 x, and their cost is given by the function C(x)=2390+2.4 x^2. Use this function to answer the following questions. Write a function, P(x), that represents the company's profit from selling x items. P(x)= Identify the vertical intercept of P(x). Write it as an ordered pair and interpret its meaning in a complete sentence. Vertical Intercept: (0,-2390) If the company sells items, they will lose How many items must be sold in order to maximize the profit? To maximize profits, 22620.42 items must be sold. Round to the nearest whole number. What is the maximum profit that can be earned? Round to the nearest cent.

Evaluating and Solving Quadratic Functions
It is very important in business for companies to determine their profit in different situations. Profit is revenue (money earned) minus costs (money spent). A company's revenue earned from selling x items is given by the function R(x)=490 x, and their cost is given by the function C(x)=2390+2.4 x^2. Use this function to answer the following questions.
Write a function, P(x), that represents the company's profit from selling x items.
P(x)=
Identify the vertical intercept of P(x). Write it as an ordered pair and interpret its meaning in a complete sentence.

Vertical Intercept: (0,-2390)

If the company sells items, they will lose
How many items must be sold in order to maximize the profit?

To maximize profits, 22620.42 items must be sold. Round to the nearest whole number.
What is the maximum profit that can be earned? Round to the nearest cent.

Transcript text: Evaluating and Solving Quadratic Functions It is very important in business for companies to determine their profit in different situations. Profit is revenue (money earned) minus costs (money spent). A company's revenue earned from selling x items is given by the function $R(x)=490 x$, and their cost is given by the function $C(x)=2390+2.4 x^{2}$. Use this function to answer the following questions. Write a function, $\mathrm{P}(\mathrm{x})$, that represents the company's profit from selling x items. \[ P(x)= \] Identify the vertical intercept of $\mathrm{P}(\mathrm{x})$. Write it as an ordered pair and interpret its meaning in a complete sentence. Vertical Intercept: $(0,-2390)$ If the company sells $\square$ items, they will lose \$ $\square$ How many items must be sold in order to maximize the profit? To maximize profits, 22620.42 items must be sold. Round to the nearest whole number. What is the maximum profit that can be earned? Round to the nearest cent.

Solution

Solution Steps

To solve the given problem, we need to follow these steps:

Define the profit function $ P(x) $: The profit function is the revenue function minus the cost function.
Identify the vertical intercept: This is the value of $ P(x) $ when $ x = 0 $.
Determine the number of items to maximize profit: This involves finding the vertex of the quadratic profit function.
Calculate the maximum profit: Substitute the number of items that maximize profit back into the profit function.

Step 1: Define the Profit Function

The profit function $ P(x) $ is defined as the difference between revenue $ R(x) $ and cost $ C(x) $: \[ P(x) = R(x) - C(x) = 490x - (2390 + 2.4x^2) = -2.4x^2 + 490x - 2390 \]

Step 2: Identify the Vertical Intercept

The vertical intercept occurs when $ x = 0 $: \[ P(0) = -2390 \] Thus, the vertical intercept is $ (0, -2390) $. This means that if no items are sold, the company incurs a loss of \$2390.

Step 3: Determine the Number of Items to Maximize Profit

To find the number of items that maximizes profit, we use the vertex formula for a quadratic function $ ax^2 + bx + c $: \[ x = -\frac{b}{2a} \] Here, $ a = -2.4 $ and $ b = 490 $: \[ x_{\text{max}} = -\frac{490}{2 \times -2.4} = 102.0833 \] Rounding to the nearest whole number, the number of items to maximize profit is $ 102 $.

Step 4: Calculate the Maximum Profit

Substituting $ x = 102 $ back into the profit function: \[ P(102) = -2.4(102)^2 + 490(102) - 2390 = 22620.40 \] Thus, the maximum profit that can be earned is \$22620.40.