You buy a new race horse for $100,000. The horse decreases in worth by 18% every year. When will the horse have a value of $5000? Show your work using the Equation Editor tool. Paragraph V Add a File B I UVA = Record Video V + v 11.
You buy a new race horse for $100,000. The horse decreases in worth by 18% every year. When will the horse have a value of $5000? Show your work using the Equation Editor tool. Paragraph V Add a File B I UVA = Record Video V + v 11.
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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![**Exercise: Depreciating Value of a Race Horse**
**Problem Statement:**
You buy a new race horse for $100,000. The horse decreases in worth by 18% every year. When will the horse have a value of $5,000? Show your work using the Equation Editor tool.
**Instructions:**
1. Utilize the Equation Editor tool provided to display your solution.
2. Clearly show each step taken to solve the problem.
**Analysis:**
Given:
- Initial value of the horse \( P \) = $100,000
- Depreciation rate \( r \) = 18% per year
- Desired value \( A \) = $5,000
We need to determine the number of years \( t \) it will take for the value of the horse to decrease to $5,000.
The value of the horse depreciates each year, which can be represented by the formula for exponential decay:
\[ A = P (1 - r)^t \]
Substitute the known values into the formula:
\[ 5000 = 100000 (1 - 0.18)^t \]
Simplify inside the parenthesis:
\[ 5000 = 100000 (0.82)^t \]
Divide both sides by 100,000:
\[ \frac{5000}{100000} = (0.82)^t \]
\[ 0.05 = (0.82)^t \]
To solve for \( t \), take the natural logarithm (ln) of both sides:
\[ \ln(0.05) = \ln((0.82)^t) \]
Use the property of logarithms that \( \ln(a^b) = b \cdot \ln(a) \):
\[ \ln(0.05) = t \cdot \ln(0.82) \]
Isolate \( t \):
\[ t = \frac{\ln(0.05)}{\ln(0.82)} \]
Using a calculator to find the values of the natural logarithms:
\[ \ln(0.05) \approx -2.9957 \]
\[ \ln(0.82) \approx -0.1983 \]
Substitute these values back into the equation for \( t \):
\[ t = \frac{-2.9957}{-0.1983} \]
\[ t \approx 15.1 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2Fca32b7aa-3965-4e51-a6b2-7db69598ed36%2Fmjw42fj_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise: Depreciating Value of a Race Horse**
**Problem Statement:**
You buy a new race horse for $100,000. The horse decreases in worth by 18% every year. When will the horse have a value of $5,000? Show your work using the Equation Editor tool.
**Instructions:**
1. Utilize the Equation Editor tool provided to display your solution.
2. Clearly show each step taken to solve the problem.
**Analysis:**
Given:
- Initial value of the horse \( P \) = $100,000
- Depreciation rate \( r \) = 18% per year
- Desired value \( A \) = $5,000
We need to determine the number of years \( t \) it will take for the value of the horse to decrease to $5,000.
The value of the horse depreciates each year, which can be represented by the formula for exponential decay:
\[ A = P (1 - r)^t \]
Substitute the known values into the formula:
\[ 5000 = 100000 (1 - 0.18)^t \]
Simplify inside the parenthesis:
\[ 5000 = 100000 (0.82)^t \]
Divide both sides by 100,000:
\[ \frac{5000}{100000} = (0.82)^t \]
\[ 0.05 = (0.82)^t \]
To solve for \( t \), take the natural logarithm (ln) of both sides:
\[ \ln(0.05) = \ln((0.82)^t) \]
Use the property of logarithms that \( \ln(a^b) = b \cdot \ln(a) \):
\[ \ln(0.05) = t \cdot \ln(0.82) \]
Isolate \( t \):
\[ t = \frac{\ln(0.05)}{\ln(0.82)} \]
Using a calculator to find the values of the natural logarithms:
\[ \ln(0.05) \approx -2.9957 \]
\[ \ln(0.82) \approx -0.1983 \]
Substitute these values back into the equation for \( t \):
\[ t = \frac{-2.9957}{-0.1983} \]
\[ t \approx 15.1 \]
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