Let's break down how to find the cost of the car 2 years from now given the current price and the annual increase rate.
1. The current price of the car is [tex]$25,000.
2. The annual increase rate is 5%, which can be expressed as a multiplier of 1.05.
We use the formula for compound interest to calculate the future cost $[/tex]C[tex]$:
\[ C = P \times (1 + r)^n \]
where:
- \( P \) is the current price ($[/tex]25,000),
- [tex]\( r \)[/tex] is the annual increase rate (0.05),
- [tex]\( n \)[/tex] is the number of years (2).
Substituting the given values into the formula, we have:
[tex]\[ C = 25,000 \times (1 + 0.05)^2 \][/tex]
[tex]\[ C = 25,000 \times (1.05)^2 \][/tex]
Calculating this:
[tex]\[ C = 25,000 \times 1.1025 \][/tex]
[tex]\[ C = 27,562.50 \][/tex]
Among the provided options, this corresponds to:
a. [tex]\( C = 25,000 \times (1.05)^2 \)[/tex]
Other options do not correctly calculate the cost of the car:
- Option 'b' [tex]\( C = 25,000 \times (1.05) \times 2 \)[/tex] results in:
[tex]\( C = 25,000 \times 1.05 \times 2 = 25,000 \times 2.10 = 52,500.00 \)[/tex]
- Option 'c' [tex]\( C = \frac{25,000 \times 2}{1.05} \)[/tex] results in:
[tex]\( C = \frac{50,000}{1.05} \approx 47,619.05 \)[/tex]
- Option 'd' [tex]\( C = 25,000 \times (2)^{105} \)[/tex] results in an astronomically large and unrealistic number.
Therefore, the best answer from the choices provided is:
[tex]\[ \boxed{a. \, C = 25,000 \times (1.05)^2} \][/tex]
Specify the cost of \[tex]$27,562.50 after 2 years for the current price of \$[/tex]25,000.
