The scrap value of a machine at the end of its useful life is given by ​S(n)equals​C(1minus​r)Superscript n​, where C is the original​ cost, n is the useful life of the machine in​ years, and r is the constant annual percentage of value lost. Find the scrap value of the following machine.
Original​ cost, ​$46 comma 000​; ​life, 6 ​years; annual rate of value​ lost, 13​%

To find the scrap value of the machine after 6 years, we can use the given formula:

\[
S(n) = C(1 - r)^n
\]

Where:
- $C$ is the original cost of the machine.
- $r$ is the annual rate of value lost (expressed as a decimal).
- $n$ is the useful life of the machine in years.

Given:
- $C = 46,000$ dollars
- $n = 6$ years
- $r = 13\%$ or 0.13 (as a decimal)

Now, plug these values into the formula:

\[
S(6) = 46,000 \times (1 - 0.13)^6
\]

First, calculate $1 - 0.13$:

\[
1 - 0.13 = 0.87
\]

Next, calculate $0.87^6$:

\[
0.87^6 \approx 0.522
\]

Finally, multiply by the original cost $C$:

\[
S(6) = 46,000 \times 0.522 \approx 24,012
\]

So, the scrap value of the machine after 6 years is approximately **$24,012**.

Would you like more details or have any questions?

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Here are 5 related questions to explore further:

1. How would the scrap value change if the annual rate of value lost was 10% instead of 13%?
2. What is the impact on the scrap value if the useful life is extended to 8 years?
3. Can you calculate the scrap value if the original cost was $50,000 and the annual rate of value lost was 15%?
4. How would the formula change if the value loss was not constant but increased each year?
5. What is the difference in scrap value between a 5-year and a 6-year useful life with the same rate of value loss?

**Tip:** Remember to always convert percentages to decimals when using them in formulas. For example, 13% becomes 0.13.

Learn how to calculate the scrap value of a machine with an original cost of $46,000, a useful life of 6 years, and an annual rate of value loss of 13%. Understand the exponential decay formula and perform calculations step-by-step. Suitable for high school students and anyone interested in financial calculations.

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