### Problem 4: Understanding Exponential DecayWhat if we started with the house having a size of 1.5 ft.? If a click to go smaller reduces the size by 50%, what formula could we use that is well connected to the first one?Given formula:\[ f(x) = a(c)^x \]Where:- $ a $ is the initial size of the house.- $ c $ is the reduction factor.- $ x $ represents the number of clicks.Based on the problem:- $ a $ is ________ and its value is $ a = 1.5 $ ft.- $ c $ is ________ and its value is $ c = 0.5 $ (since the size reduces by 50% each click).- $ x $ represents ________ the number of clicks.This formula helps us understand how the size of the house changes exponentially with each click.**Note:** This problem involves understanding how exponential decay works, which is a crucial concept in mathematics, especially in contexts such as population decline, radioactive decay, and in this case, size reduction. ### Stock Growth and Decline Formulas#### 5. Stock Growth FormulaIf you own one share of stock in GameStop that was worth $100 when you bought it, what formula would show it having 4% growth a year?The formula given is:\[ f(x) = a(c)^x \]To understand the components of this formula:- **a** is __________________ and its value is $ a = $ _______- **c** is __________________ and its value is $ c = $ _______- **x** represents __________________#### 6. Stock Decline FormulaWhat if your hundred dollars share instead lost 4% per year? What does this formula look like?The formula remains similar:\[ f(x) = a(c)^x \]We need to define the components for the decline scenario:- **a** is __________________ and its value is $ a = $ _______- **c** is __________________ and its value is $ c = $ _______- **x** represents __________________### Biological Half-Life of Medication#### 7. Over-the-Counter Headache RelieverThe biological half-life of an over-the-counter headache reliever is 1 hour. This means that the amount of pain reliever in the body is reduced by 50% every hour. Suppose Jasper took 200 mg of this medication. Write a function $ f $ that you can use to model t hours after taking the medication.

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4: What if we started with the house having a size of 1.5 ft.? If a click to go smaller reduces the size by 50%, what formula could we use that is well connected to the first one? f(x) = a(c) a is and its value is a = c is and its value is c = x represents

4: What if we started with the house having a size of 1.5 ft.? If a click to go smaller reduces the size by 50%, what formula could we use that is well connected to the first one? f(x) = a(c) a is and its value is a = c is and its value is c = x represents

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Problem 4: Understanding Exponential Decay

What if we started with the house having a size of 1.5 ft.? If a click to go smaller reduces the size by 50%, what formula could we use that is well connected to the first one?

Given formula:
\[ f(x) = a(c)^x \]

Where:
- $ a $ is the initial size of the house.
- $ c $ is the reduction factor.
- $ x $ represents the number of clicks.

Based on the problem:
- $ a $ is ________ and its value is $ a = 1.5 $ ft.
- $ c $ is ________ and its value is $ c = 0.5 $ (since the size reduces by 50% each click).
- $ x $ represents ________ the number of clicks.

This formula helps us understand how the size of the house changes exponentially with each click.

**Note:** This problem involves understanding how exponential decay works, which is a crucial concept in mathematics, especially in contexts such as population decline, radioactive decay, and in this case, size reduction.

### Stock Growth and Decline Formulas

#### 5. Stock Growth Formula

If you own one share of stock in GameStop that was worth $100 when you bought it, what formula would show it having 4% growth a year?

The formula given is:
\[ f(x) = a(c)^x \]

To understand the components of this formula:
- **a** is __________________ and its value is $ a = $ _______
- **c** is __________________ and its value is $ c = $ _______
- **x** represents __________________

#### 6. Stock Decline Formula

What if your hundred dollars share instead lost 4% per year? What does this formula look like?

The formula remains similar:
\[ f(x) = a(c)^x \]

We need to define the components for the decline scenario:
- **a** is __________________ and its value is $ a = $ _______
- **c** is __________________ and its value is $ c = $ _______
- **x** represents __________________

### Biological Half-Life of Medication

#### 7. Over-the-Counter Headache Reliever

The biological half-life of an over-the-counter headache reliever is 1 hour. This means that the amount of pain reliever in the body is reduced by 50% every hour. Suppose Jasper took 200 mg of this medication. Write a function $ f $ that you can use to model t hours after taking the medication.

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