Solve and show work. 2) The amount of hairs on Mr. Mahers head has been decreasing exponentially at a rate of 5. 5% per year since 1998 when he had 110,000 hairs. a) Write an equation that models hair where t is the number of years since 1998. b) How many hairs on his head today. * (2022) ?
Solve and show work. 2) The amount of hairs on Mr. Mahers head has been decreasing exponentially at a rate of 5. 5% per year since 1998 when he had 110,000 hairs. a) Write an equation that models hair where t is the number of years since 1998. b) How many hairs on his head today. * (2022) ?
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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![**Mathematics Problem Solving**
**Problem 2:**
The amount of hairs on Mr. Maher's head has been decreasing exponentially at a rate of 5.5% per year since 1998 when he had 110,000 hairs.
**a) Write an equation that models hair, where \( t \) is the number of years since 1998.**
**b) How many hairs on his head today (2022)?**
**Solution:**
To model the number of hairs using exponential decay, we use the formula:
\[ H(t) = H_0 \times (1 - r)^t \]
Where:
- \( H(t) \) = the number of hairs after \( t \) years,
- \( H_0 \) = the initial number of hairs,
- \( r \) = the decay rate,
- \( t \) = the number of years since the initial count.
Given:
- \( H_0 = 110,000 \) hairs,
- \( r = 0.055 \) (since 5.5% = 0.055),
- Today is 2022, hence \( t = 2022 - 1998 \).
Now substitute the values into the equation to find \( H(t) \).
For part (b),
\[ H(2022 - 1998) = H(24) \]
Calculate to find \( H(24) \).
This problem helps us understand how exponential functions model real-world scenarios, such as decay or growth over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76555388-6a8d-4d01-aabe-a551666898f4%2F05c664b4-dd53-41bf-94fe-3f6859b0a6b7%2F412h24r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem Solving**
**Problem 2:**
The amount of hairs on Mr. Maher's head has been decreasing exponentially at a rate of 5.5% per year since 1998 when he had 110,000 hairs.
**a) Write an equation that models hair, where \( t \) is the number of years since 1998.**
**b) How many hairs on his head today (2022)?**
**Solution:**
To model the number of hairs using exponential decay, we use the formula:
\[ H(t) = H_0 \times (1 - r)^t \]
Where:
- \( H(t) \) = the number of hairs after \( t \) years,
- \( H_0 \) = the initial number of hairs,
- \( r \) = the decay rate,
- \( t \) = the number of years since the initial count.
Given:
- \( H_0 = 110,000 \) hairs,
- \( r = 0.055 \) (since 5.5% = 0.055),
- Today is 2022, hence \( t = 2022 - 1998 \).
Now substitute the values into the equation to find \( H(t) \).
For part (b),
\[ H(2022 - 1998) = H(24) \]
Calculate to find \( H(24) \).
This problem helps us understand how exponential functions model real-world scenarios, such as decay or growth over time.
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