Question
A machine depreciates by one fourth
of its value each year. If it costs $590 comma 000 new, what is its value after 6 years?
.
.
.
Question content area right
Part 1
After 6 years the machine is worth $
enter your response here.
(Simplify your answer. Round to the nearest dollar if needed.)
of its value each year. If it costs $590 comma 000 new, what is its value after 6 years?
.
.
.
Question content area right
Part 1
After 6 years the machine is worth $
enter your response here.
(Simplify your answer. Round to the nearest dollar if needed.)
Answers
GPT-4o mini
To find the value of the machine after 6 years, we start with its initial cost of $590,000 and calculate its value at the end of each year, considering that it depreciates by one fourth (25%) each year.
This means that at the end of each year, the machine retains \(\frac{3}{4}\) (or 75%) of its value from the previous year.
The value of the machine after \(n\) years can be calculated using the formula:
\[
\text{Value after } n \text{ years} = \text{Initial Value} \times \left(\frac{3}{4}\right)^n
\]
In this case:
- Initial Value = $590,000
- \(n = 6\)
Now we can calculate the value after 6 years:
\[
\text{Value after 6 years} = 590,000 \times \left(\frac{3}{4}\right)^6
\]
Calculating \(\left(\frac{3}{4}\right)^6\):
\[
\left(\frac{3}{4}\right)^6 = \frac{729}{4096} \approx 0.177978515625
\]
Now plug this into the value formula:
\[
\text{Value after 6 years} = 590,000 \times 0.177978515625 \approx 104,691.59
\]
Rounding to the nearest dollar, the value of the machine after 6 years is:
\[
\text{Value after 6 years} \approx 104,692
\]
So, the machine is worth **$104,692** after 6 years.
This means that at the end of each year, the machine retains \(\frac{3}{4}\) (or 75%) of its value from the previous year.
The value of the machine after \(n\) years can be calculated using the formula:
\[
\text{Value after } n \text{ years} = \text{Initial Value} \times \left(\frac{3}{4}\right)^n
\]
In this case:
- Initial Value = $590,000
- \(n = 6\)
Now we can calculate the value after 6 years:
\[
\text{Value after 6 years} = 590,000 \times \left(\frac{3}{4}\right)^6
\]
Calculating \(\left(\frac{3}{4}\right)^6\):
\[
\left(\frac{3}{4}\right)^6 = \frac{729}{4096} \approx 0.177978515625
\]
Now plug this into the value formula:
\[
\text{Value after 6 years} = 590,000 \times 0.177978515625 \approx 104,691.59
\]
Rounding to the nearest dollar, the value of the machine after 6 years is:
\[
\text{Value after 6 years} \approx 104,692
\]
So, the machine is worth **$104,692** after 6 years.
wrong
GPT-4o mini
Let’s carefully recalculate the value of the machine after 6 years, given that it depreciates by one fourth (25%) each year.
### Initial Value
- Initial cost of the machine: **$590,000**
### Depreciation
As the machine depreciates by one fourth each year, it retains \( \frac{3}{4} \) of its value each year.
### Formula
The value after \( n \) years can be calculated with the formula:
\[
\text{Value after } n \text{ years} = \text{Initial Value} \times \left(\frac{3}{4}\right)^n
\]
### Calculation for 6 Years
1. **Calculate \( \left(\frac{3}{4}\right)^6 \):**
\[
\left(\frac{3}{4}\right)^6 = \frac{729}{4096} \approx 0.177978515625
\]
2. **Multiply by the initial value:**
\[
\text{Value after 6 years} = 590,000 \times 0.177978515625 \approx 104,691.59
\]
3. **Round to the nearest dollar:**
\[
\text{Value after 6 years} \approx 104,692
\]
This calculation confirms that the value of the machine after 6 years remains approximately **$104,692**.
If this answer is incorrect in a specific context or you're looking for a different approach or formula, please provide more details so I can assist you accurately.
### Initial Value
- Initial cost of the machine: **$590,000**
### Depreciation
As the machine depreciates by one fourth each year, it retains \( \frac{3}{4} \) of its value each year.
### Formula
The value after \( n \) years can be calculated with the formula:
\[
\text{Value after } n \text{ years} = \text{Initial Value} \times \left(\frac{3}{4}\right)^n
\]
### Calculation for 6 Years
1. **Calculate \( \left(\frac{3}{4}\right)^6 \):**
\[
\left(\frac{3}{4}\right)^6 = \frac{729}{4096} \approx 0.177978515625
\]
2. **Multiply by the initial value:**
\[
\text{Value after 6 years} = 590,000 \times 0.177978515625 \approx 104,691.59
\]
3. **Round to the nearest dollar:**
\[
\text{Value after 6 years} \approx 104,692
\]
This calculation confirms that the value of the machine after 6 years remains approximately **$104,692**.
If this answer is incorrect in a specific context or you're looking for a different approach or formula, please provide more details so I can assist you accurately.