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Answer :
Let's find out how many years it will take for the furniture valued at [tex]$10,000 to depreciate to a value of $[/tex]6,000 at a constant rate of 7% every 2 years. We'll do this by following these steps:
1. Understand the Depreciation Model:
The model for depreciation given is:
[tex]\( \text{final value} = \text{initial value} \times (1 - \text{depreciation rate})^{t/2} \)[/tex]
In this case, the initial value is [tex]$10,000, and it depreciates by 7% every 2 years. Thus, the depreciation rate is \(0.07\), so after each two-year period, the furniture retains \(93\%\) of its value, or \(0.93\) times its previous value.
2. Set Up the Equation:
We need to find when the initial value of $[/tex]10,000 becomes [tex]$6,000. Substituting the values into the formula, we have:
\( 6000 = 10000 \times (0.93)^{t/2} \)
3. Solve for Time \(t\):
- Divide both sides by $[/tex]10,000 to simplify the equation:
[tex]\( \frac{6000}{10000} = (0.93)^{t/2} \)[/tex]
- Simplify the fraction:
[tex]\( 0.6 = (0.93)^{t/2} \)[/tex]
4. Use Logarithms to Solve for [tex]\(t\)[/tex]:
- Take the natural logarithm (ln) of both sides to isolate [tex]\(t\)[/tex]:
[tex]\[ \ln(0.6) = \frac{t}{2} \times \ln(0.93) \][/tex]
- Solve for [tex]\(t\)[/tex]:
[tex]\[ t = \frac{2 \times \ln(0.6)}{\ln(0.93)} \][/tex]
5. Numerical Solution:
After calculating this expression, we find that [tex]\(t \approx 14.08\)[/tex].
So, it will take approximately 14.08 years for the furniture to depreciate from [tex]$10,000 to $[/tex]6,000.
1. Understand the Depreciation Model:
The model for depreciation given is:
[tex]\( \text{final value} = \text{initial value} \times (1 - \text{depreciation rate})^{t/2} \)[/tex]
In this case, the initial value is [tex]$10,000, and it depreciates by 7% every 2 years. Thus, the depreciation rate is \(0.07\), so after each two-year period, the furniture retains \(93\%\) of its value, or \(0.93\) times its previous value.
2. Set Up the Equation:
We need to find when the initial value of $[/tex]10,000 becomes [tex]$6,000. Substituting the values into the formula, we have:
\( 6000 = 10000 \times (0.93)^{t/2} \)
3. Solve for Time \(t\):
- Divide both sides by $[/tex]10,000 to simplify the equation:
[tex]\( \frac{6000}{10000} = (0.93)^{t/2} \)[/tex]
- Simplify the fraction:
[tex]\( 0.6 = (0.93)^{t/2} \)[/tex]
4. Use Logarithms to Solve for [tex]\(t\)[/tex]:
- Take the natural logarithm (ln) of both sides to isolate [tex]\(t\)[/tex]:
[tex]\[ \ln(0.6) = \frac{t}{2} \times \ln(0.93) \][/tex]
- Solve for [tex]\(t\)[/tex]:
[tex]\[ t = \frac{2 \times \ln(0.6)}{\ln(0.93)} \][/tex]
5. Numerical Solution:
After calculating this expression, we find that [tex]\(t \approx 14.08\)[/tex].
So, it will take approximately 14.08 years for the furniture to depreciate from [tex]$10,000 to $[/tex]6,000.
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