In 1997, researchers at Texas A&M University estimated the operating costs of cotton gin plans of various sizes. A quadratic model of cost (in thousands of dollars) for the largest plants was found to be very similar to: C(q) 0.023q + 26.4q + 338 where q is the annual quanity of bales (in thousands) produced by the plant. Revenue was estimated at $65 per bale of cotton. Find the following (but be cautious and play close attention to the units): A) The Marginal Cost function: MC(q) = B) The Marginal Revenue function: MR(q)
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.
![### Cost Estimation of Cotton Gin Plants
In 1997, researchers at Texas A&M University estimated the operating costs of cotton gin plants of various sizes. A quadratic model of cost (in thousands of dollars) for the largest plants was found to be very similar to:
\[
C(q) = 0.023q^2 + 26.4q + 338
\]
where \( q \) is the annual quantity of bales (in thousands) produced by the plant. Revenue was estimated at $65 per bale of cotton.
### Tasks
Find the following (but be cautious and play close attention to the units):
#### A) The Marginal Cost function:
\[
MC(q) = \underline{\hspace{3cm}}
\]
#### B) The Marginal Revenue function:
\[
MR(q) = \underline{\hspace{3cm}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6b4032-4723-40f9-bafe-2ad74429fe12%2F8060439f-8dc9-42cf-97b0-7d8df4741a64%2Fpor9yta_processed.jpeg&w=3840&q=75)
![### Marginal Profit Analysis
#### C) The Marginal Profit function:
\[ MP(q) = \_\_\_\_\_\_\_\_\_\_\_\_\_ \]
#### D) The Marginal Profits for \( q = 419 \) thousand units:
\[ MP(419) = \_\_\_\_\_\_\_\_\_\_\_\_\_ \quad (\text{see Part E for units}) \]
#### Which of the following represent the proper units for the answer to Part D?
- [ ] units
- [ ] thousands of units per dollar
- [ ] thousands of dollars per unit
- [ ] dollars per unit
- [ ] dollars
- [ ] units per dollar
### Explanation of Marginal Profit Function:
The marginal profit function, denoted by \( MP(q) \), represents the additional profit gained from selling one more unit of a product when producing \( q \) units. It is a crucial concept in economics and business as it helps in understanding the profitability of producing additional units.
For further learning, ensure to solve Part E to identify the correct units of the marginal profit function. This knowledge supports making informed business decisions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6b4032-4723-40f9-bafe-2ad74429fe12%2F8060439f-8dc9-42cf-97b0-7d8df4741a64%2F2lqm03d_processed.jpeg&w=3840&q=75)
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