To determine the inflation-adjusted cost of a boat that has an initial cost of \[tex]$210,000 today, given an average annual inflation rate of 3.5% over the next 5 years, we will use the function:
\[ C(t) = C_0 (1 + r)^t \]
where:
- \( C(t) \) is the cost of the boat after \( t \) years,
- \( C_0 \) is the initial cost of the boat (today's cost),
- \( r \) is the average annual inflation rate,
- \( t \) is the number of years.
Step-by-Step Solution:
1. Identify the given values:
- Initial cost, \( C_0 \) = \$[/tex]210,000
- Annual inflation rate, [tex]\( r \)[/tex] = 3.5% = 0.035 (as a decimal)
- Number of years, [tex]\( t \)[/tex] = 5
2. Substitute the given values into the formula:
[tex]\[ C(t) = 210,000 \times (1 + 0.035)^5 \][/tex]
[tex]\[ C(t) = 210,000 \times (1.035)^5 \][/tex]
3. Calculate [tex]\( (1.035)^5 \)[/tex]:
[tex]\[ (1.035)^5 \approx 1.1877 \][/tex]
4. Multiply the initial cost [tex]\( C_0 \)[/tex] by the result from the previous step:
[tex]\[ C(t) = 210,000 \times 1.1877 \][/tex]
[tex]\[ C(t) \approx 249,414.12418584366 \][/tex]
5. Round the result to two decimal places:
[tex]\[ C(t) \approx 249,414.12 \][/tex]
So, the inflation-adjusted cost of the \[tex]$210,000 boat in 5 years, with an annual inflation rate of 3.5%, will be approximately \$[/tex]249,414.12.
