**Modeling Profit Growth Over Time**A company that sells woollen hats has seen a generally linear growth in their profits over 2019, though there is also a seasonal impact. The data from 2019 was used to create a model for the profit per month $ P $ as a function of the number of months since the beginning of 2019:\[ P = f(m) = 1,000m + \frac{12,000}{\pi} \cos \left( \frac{\pi}{6} m \right). \]**Tasks:**Use the equation above to find the following, explaining your approach, and showing and explaining your computations.**(a) The minimum profit in 2019.****(b) A prediction for the maximum profit in 2020.****Explanation of the Equation:**1. **Linear Growth Component:** $ 1,000m $ - This term represents the linear increase in profit as a function of the number of months $ m $.2. **Seasonal Impact Component:** $ \frac{12,000}{\pi} \cos \left( \frac{\pi}{6} m \right) $ - This term models the seasonal fluctuations in profit. The cosine function introduces periodic variations, indicating consistent seasonal highs and lows.**Approach to Solving the Tasks:**For task (a), finding the minimum profit in 2019 involves identifying the points where the seasonal component minimizes the overall profit function. Similarly, for task (b), determining the maximum profit for 2020 involves identifying the points where the seasonal impact maximizes the profit trend, extended into the future period.

Let m represents the number of months since the beginning of 2019 and P represents the profit. The…

A company that sells woollen hats has seen a generally linear growth in their profits over 2019, though there is also a seasonal impact. The data from 2019 was used to create a model for the profit per month P as a function of the number of months since the beginning of 2019, Р3 f(m) — 1,000 т + 12, 000 CoS :(). Use the equation above to find the following, explaining your approach, and showing and explaining your computations. (a) The minimum profit in 2019. (b) A prediction for the maximum profit in 2020.

A company that sells woollen hats has seen a generally linear growth in their profits over 2019, though there is also a seasonal impact. The data from 2019 was used to create a model for the profit per month P as a function of the number of months since the beginning of 2019, Р3 f(m) — 1,000 т + 12, 000 CoS :(). Use the equation above to find the following, explaining your approach, and showing and explaining your computations. (a) The minimum profit in 2019. (b) A prediction for the maximum profit in 2020.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: The probability of precipitation in a California city varies from a peak of 0.38 (38%) in January to…

Q: The population of the world is given approximately by f(t) = 7.38e0.0119t billion people, where t is…

A: Use the help of derivative and solve the problem.

Q: Suppose a certain variable star has a period of 10 days. The average brightness of the star is 3.1…

A: According to the given problem, find the equation of y:

Q: The rate of increase of the number of working married couples M (in millions) in the United States…

A: Given: The rate of increase of the number of working married couples M in the US from 2008 through…

Q: In 1994, the moose population in a park was measured to be 3500. By 1999, the population was…

Q: National spending on health has increased approximately exponentially from $0.7 trillion in 1990 to…

A: Find the equation of f(t) and determine t for the given conditions.

Q: When you are 18 years old you have a hair that is 16 centimeters long, and your hair grows about 12…

Q: the price "P (t)" of a certain model of technological equipment in dollars ($) is given by (t) =…

Q: population of a town has been growing linearly since 2000. The population t years since 2000 can be…

A: The population of a town has been growing linearly since 2000. The population t years since 2000 can…

Q: he projected world population (in billions of people) t years after 2000 is modeled by the function…

A: To find: The value of r if the expected population for the year 2012 is 7 billion. Given: The…

Q: In 1994, the moose population in a park was measured to be 4420. By 1996, the population was…

Q: On August 4, 2007, NASA's Phoenix Mars Lander was launched into space to search for life in the icy…

A: The distance that the tank falls in t second is s(t)=16t2 (a) The average speed of the tank during…

Q: The gross national economy of a certain country (in billions of U.S. dollars) can be approximated by…

A: Given, The gross national economy of a certain country can be approximately by P(t) = 1033e0.178t…

Q: In 1992, the moose population in a park was measured to be 4070. By 1998, the population was…

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**Modeling Profit Growth Over Time**

A company that sells woollen hats has seen a generally linear growth in their profits over 2019, though there is also a seasonal impact. The data from 2019 was used to create a model for the profit per month $ P $ as a function of the number of months since the beginning of 2019:

\[ P = f(m) = 1,000m + \frac{12,000}{\pi} \cos \left( \frac{\pi}{6} m \right). \]

**Tasks:**

Use the equation above to find the following, explaining your approach, and showing and explaining your computations.

**(a) The minimum profit in 2019.**

**(b) A prediction for the maximum profit in 2020.**

**Explanation of the Equation:**

1. **Linear Growth Component:** $ 1,000m $
- This term represents the linear increase in profit as a function of the number of months $ m $.

2. **Seasonal Impact Component:** $ \frac{12,000}{\pi} \cos \left( \frac{\pi}{6} m \right) $
- This term models the seasonal fluctuations in profit. The cosine function introduces periodic variations, indicating consistent seasonal highs and lows.

**Approach to Solving the Tasks:**

For task (a), finding the minimum profit in 2019 involves identifying the points where the seasonal component minimizes the overall profit function. Similarly, for task (b), determining the maximum profit for 2020 involves identifying the points where the seasonal impact maximizes the profit trend, extended into the future period.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

In 1993, the moose population in a park was measured to be 3200. By 1996, the population was measured again to be 2100. If the population continues to change linearly: Find a formula for yhe moose pupulation, P,in terms of t, the years since 1990. P(t)= what does your model predict the moose population to be in 2009?
Weekly sales of graphing calculators can be modeled by the equation s(t) = 10 – te-t/20 - where s is the number of graphing calculators sold per week after t weeks. How many graphing calculators will be sold in the first 80 weeks? calculators (include two decimal places)
Having completed your first few tasks, you move now to a portion of the lab where important bacteria are being grown. Based on the data collected, you have determined that the population of this bacteria in the range measured can be modeled as A (t) = 195(1.31)t, where t is given in days. To the nearest whole number, what will the bacteria population be after 5 days? The bacteria population will be Number
3. The population of wolves on an island can be modeled by the equation P(t) = 120e-698t where t is the number of years since 2000. %3D a. What was the population in the year 2000? b. Describe what is happening to the population. Is it growing or shrinking? c. When will there be only 20 wolves on the island?
When you are 18 years old you have a hair that is 15 centimeters long, and your hair grows about 12 centimeters each year. Let H(t) be the length, in centimeters, of that hair t years after age 18. (a) Find a formula that gives H as a linear function of t. H= (b) How long will it take for the hair to reach a length of 100 centimeters? (Round your answer to two decimal places.)
A mobile phone company records their annual sales on 31 December every year. Paul thinks that the annual sales, S million, can be modelled by the equation S= ab', where a ar b are both positive constants and t is the number of years since 31st December 2015. Paul tests his theory by using the annual sales figures from 31t December 2015 to 31st Decemb 2019. He plots these results on a graph, with t on the horizontal axis and log,,S on the vertic axis. (a) Explain why, if Paul's model is correct, the results should lie on a straight line of best fit on h graph. The results lie on a straight line of best fit which has a gradient of 0.146 and an intercept on vertical axis of 0.583. (b) Use these values to obtain estimates for a and b, correct to 2 significant figures. (c) Use this model to predict the year in which, on the 31st December, the annual sales wvould fi be recorded as greater than 200 million. (d) Give a reason why this prediction may not be reliable.