I don’t understand this Calculus question and need help to study.

Suppose in one year, total revenues from digital sales of pop/rock, tropical (salsa/merengue/cumbia/bachata), and urban (reggaeton) Latin music in a certain country amounted to $27 million. Pop/rock music brought in twice as much as the other two categories combined and $11 million more than tropical music. How much revenue was earned from digital sales in each of the three categories?

pop/rock music$ million

tropical music$ million

urban Latin music$ million

The B&K Real Estate Company sells homes and is currently serving the Southeast region. It has recently expanded to cover the Northeast states. The B&K realtors are excited to now cover the entire East Coast and are working to prepare their southern agents to expand their reach to the Northeast.

B&K has hired your company to analyze the Northeast home listing prices in order to give information to their agents about the mean listing price at 95% confidence. Your company offers three analysis packages: one based on a sample size of 100 listings, one based on 1,000 listings, and another based on a sample size of 4,000 listings. Because there is an additional cost for data collection, your company charges more for the package with 4,000 listings than for the package with 100 listings.

Bronze Package – Sample size of 100 listings:

• 95% confidence interval for the mean of the Northeast house listing price has a margin of error of $24,500
• Cost for service to B&K: $2,000

Silver Package – Sample size of 1,000 listings:

• 95% confidence interval for the mean of the Northeast house listing price has a margin of error of $7,750
• Cost for service to B&K: $10,000

Gold Package – Sample size of 4,000 listings:

• 95% confidence interval for the mean of the Northeast house listing price has a margin of error of $3,900
• Cost for service to B&K: $25,000

The B&K management team does not understand the tradeoff between confidence level, sample size, and margin of error. B&K would like you to come back with your recommendation of the sample size that would provide the sales agents with the best understanding of northeast home prices at the lowest cost for service to B&K.

In other words, which option is preferable?

• Spending more on data collection and having a smaller margin of error
• Spending less on data collection and having a larger margin of error
• Choosing an option somewhere in the middle

For your initial post:

• Formulate a recommendation and write a confidence statement in the context of this scenario. For the purposes of writing your confidence statement, assume the sample mean house listing price is $310,000 for all packages. “I am [#] % confident the true mean . . . [in context].”
• Explain the factors that went into your recommendation, including a discussion of the margin of error

For your response posts to your peers, choose two different confidence intervals for your responses. Do you think the agents would prefer a different confidence interval than their management? What advantages and disadvantages would there be in having different confidence intervals for the agents? Explain your thought process and reasoning in your response.

Step-by-step solved example for the Composition of functions

Here are some Discussion assignment clarifications, with a solved example, for this week’s topic:
Since most of the people have questions about the composition of functions, we have to work together on this assignment. Please review section 3.4 for the composition of functions’ definitions and examples, as this subject is challenging, but interesting, and useful, and requires some very good analytical and attention to detail skills.

Note 1:

Somewhere, in the fine lines, I stated that the temperature conversion is just a starting point for our composition of functions’ analysis. The only topic that we discuss here is about the process of composition of functions which is not an easy subject. Please note that a conversion formula (C to F or F to C) is considered the function for our practice exercise.

Please see below my solved example, with detailed explanations and comments. I will choose the month of October, where the average temperature in my case is m = 63, (M or m=month).

Find: F(Fahrenheit) = F(M) = F(Oct) = F(63) = 63

C(Celsius) = C(M) = C(Oct) = C(63) = 5/9(M-32) = 5/9(63-32) = 5/9 x 31= 155/9 = 17.2

1. Calculate (C ᵒ F)(M) for the month of your choice (October), and show formula and steps:

(C ᵒ F)(M) = (C ᵒ F)(63) = C(F(63)) = C(63) = 155/9 (it is OK to repeat the steps)

2. Discuss the meaning of the function (C ∘ F)(M) = C(F(M))

A “Function Composition” is applying one function to the results of another function. In this operation the function C is applied to the result of applying the function F to M. (Please do not limit the discussion to the topic of temperature conversion only as we can have any letters/variables for our practice exercises).

Note 2:

Since we can have any letters/variables for our functions it is not important what a temperature conversion is but how do we solve a composition of functions, which can become complicated fast. In other words, how do we de-compose (break up) a function into a composition of other functions? How do we make things simpler in order to find a solution?

3. How does the composition of functions in parts 1 and 2 compare to (F ∘ C)(M)? Are they the same? or are they inverse?

a). Composition of Celsius and Fahrenheit (result in Celsius):

(C ᵒ F)(M) = (C ᵒ F)(63) = C(F(63)) = C(63) = 5/9(63-32) = 5/9×31 = 155/9

b). Composition of Fahrenheit and Celsius (result in Fahrenheit):

(F ᵒ C)(M) = (F ᵒ C)(63) = F(C(63)) = F(155/9) = 9/5C+32 = 9/5×155/9+32 = 31+32 = 63

As shown above the two functions are inverse.

Note 3:

We use the Celsius conversion formula to solve for variable F (Fahrenheit):

C = 5/9(F-32) Celsius (from Fahrenheit)

F = 9/5C + 32 Fahrenheit (from Celsius)

For C = 155/9 (easier to simplify if we keep the results as a fraction) we have:

F = 9/5×155/9 +32 = 31+32 = 63

Note 4: Final Observations

In a business setting you have a maximum of 3 or 4 minutes to present your work. This is why your comments must be concise and precise. Also, you have to be able to say a lot with very little. In terms of analytical and attention to detail skills, I think this assignment is one of the best, and an eye opener, since Algebra is the second most unforgiving type of mathematics (second only to Statistics).

I end my comment with a Composition of functions’ Summary:

• A “Function Composition” is applying one function to the results of another.
• (g º f)(x) = g(f(x)), first apply f( ), then apply g( )
• Some functions can be de-composed into two (or more) simpler functions.==============================================================================================================================================================================QUESTION
• Consider the following two functions:
  • F(M): The average temperature (F) in Fahrenheit during month (M) of the year.

  Month (M)	F (M)
  January	42
  February	43
  March	52
  April	64
  May	72
  June	83
  July	86
  August	84
  September	73
  October	65
  November	55
  December	44

  • C(F): The conversion formula to calculate the temperature in Celsius (C) based on the temperature in Fahrenheit (F).
  C(F)=59(F−32)$C\left(F\right)=\frac{5}{9}\left(F–32\right)$

  Your task for this discussion is as follows:

  1. Calculate (C ∘ F) for the month of your choice.
  2. Discuss the meaning of the function (C ∘ F)(M).
  3. Now think about the composition, (F ∘ C). Does this composition make sense? If so, how does it compare to (C ∘ F)? If not, why not?

I need an explanation for this Calculus question to help me study.

1. The definite integral of a function is always a positive quantity.
2. The area between two distinct curves is always a positive quantity.
3. The consumers’ surplus and the producers’ surplus equal each other.
4. In the trapezoidal rule, the number of subintervals must be even.
5. Simpson’s rule usually gives a better approximation than the trapezoidal rule