Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be p=190−0.5x. It also determines that the total cost of producing x suits is given by C(x)=2500+0.75x2. a) Find the total revenue, R(x). b) Find the total profit, P(x). c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![**Title: Maximizing Profit for Raggs, Ltd.**
**Overview:**
Raggs, Ltd., a clothing firm, analyzes its pricing and production strategy to maximize profit from selling suits. The firm has defined two critical functions: the price per suit and the cost of production.
**Problem Statement:**
- In order to sell \( x \) suits, the price per suit (\( p \)) must be defined by the equation:
\[
p = 190 - 0.5x
\]
- The total cost (\( C(x) \)) of producing \( x \) suits is given as:
\[
C(x) = 2500 + 0.75x^2
\]
**Tasks:**
a) **Find the total revenue, \( R(x) \):**
b) **Find the total profit, \( P(x) \):**
c) **Determine the number of suits the company must produce and sell to maximize profit:**
d) **Calculate the maximum profit:**
e) **Identify the price per suit that must be charged to maximize profit:**
**Solutions:**
- **a) Total Revenue, \( R(x) \):**
- **b) Total Profit, \( P(x) \):**
- **c) Optimal Number of Suits:**
\[
\_\_\_\_ \text{ suits}
\]
- **d) Maximum Profit:**
\[
\text{The maximum profit is } \$\_\_\_\_.
\]
- **e) Price Per Unit for Maximum Profit:**
\[
\text{The price per unit must be } \$\_\_\_\_.
\]
**Note:**
- Ensure each function is clearly defined and used to solve the optimization problem by employing calculus or other mathematical techniques.
- Completing the tasks will aid the company in strategizing its market approach, focusing on profitability and efficient production.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabec27d7-ece1-4ee9-9bb7-845ecd00cf57%2Ff80c8346-68d8-414f-82f3-d9e341531cb7%2Fx4g8da_processed.png&w=3840&q=75)
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