**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

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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.

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