https://file.aiquickdraw.com/mathbot/file/1725590451121kl7aku0f.jpg

https://file.aiquickdraw.com/mathbot/file/172559045112116t7cp6d.jpg

Answer the questions in the same order that they show up

I will analyze the image content and solve the questions based on the provided data.

Let's begin with the first part:

---

### **1. Part a: Price-Demand Function**
The price-demand function is given as:

\[ p(x) = 2000 - 60x \]

where \( p(x) \) is the price per computer in dollars when \( x \) thousand computers are sold.

### **Revenue Function**
The revenue function is given by:

\[ R(x) = p(x) \cdot x = (2000 - 60x) \cdot x \]
Expanding the function:

\[ R(x) = 2000x - 60x^2 \]

This is the revenue function in thousands of dollars.

---

#### **a) Graph the Revenue Function**
To plot the revenue function, \( R(x) = 2000x - 60x^2 \), you would plot values of \( R(x) \) for different \( x \) (number of computers in thousands).

Let's first compute some key points for the graph:
- For \( x = 0 \):
  \[
  R(0) = 2000(0) - 60(0)^2 = 0
  \]
- For \( x = 10 \):
  \[
  R(10) = 2000(10) - 60(10)^2 = 20000 - 6000 = 14000
  \]
- For \( x = 20 \):
  \[
  R(20) = 2000(20) - 60(20)^2 = 40000 - 24000 = 16000
  \]
- For \( x = 30 \):
  \[
  R(30) = 2000(30) - 60(30)^2 = 60000 - 54000 = 6000
  \]

These values will help plot a parabolic curve on the graph, with the maximum revenue between \( x = 10 \) and \( x = 30 \). The graph will rise until it hits a maximum, then decrease.

---

### **2. Part b: Cost Function**

The cost function is given by:
\[
C(x) = 4000 + 50x
\]
This is a linear function, where the fixed cost is 4000 and the variable cost increases by 50 for each additional \( x \) (in thousands of computers).

Let's calculate some values:
- For \( x = 0 \):
  \[
  C(0) = 4000
  \]
- For \( x = 10 \):
  \[
  C(10) = 4000 + 50(10) = 4500
  \]
- For \( x = 20 \):
  \[
  C(20) = 4000 + 50(20) = 5000
  \]
- For \( x = 30 \):
  \[
  C(30) = 4000 + 50(30) = 5500
  \]

These values can be used to plot the cost function.

---

#### **c) Find the Break-Even Points**

The break-even points occur when revenue equals cost, i.e., \( R(x) = C(x) \).

Set the functions equal to each other:
\[
2000x - 60x^2 = 4000 + 50x
\]
Rearranging the terms:
\[
-60x^2 + 2000x - 50x - 4000 = 0
\]
\[
-60x^2 + 1950x - 4000 = 0
\]

We can solve this quadratic equation to find the break-even points. I will calculate the roots next.

Let me calculate the break-even points.

The manufacturing research department for a company that manufactures and sells notebook computers established the following price-demand and revenue functions: p(x) = 2000 - 60x and R(x) = p(x) * x = (2000 - 60x) * x. Find break-even points, maximum profit, and analyze graphs.

This problem involves analyzing the price-demand function p(x) = 2000 - 60x and revenue function R(x) = 2000x - 60x^2 to determine break-even points, maximum profit, and graph the functions. Suitable for high school or early college students studying algebra and economics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation