Let's explore the given equation in detail to find the correct statement.
The equation given is:
[tex]\[ V(t) = 12,000 \cdot (0.75)^t \][/tex]
Here, [tex]\( V(t) \)[/tex] represents the value of the motorcycle after [tex]\( t \)[/tex] years. To understand this equation fully, we shall break it down into its components:
1. The constant term 12000 in the equation represents the initial value of the motorcycle when it was purchased, i.e., when [tex]\( t = 0 \)[/tex].
2. The term [tex]\( (0.75)^t \)[/tex] indicates how the value of the motorcycle changes over time.
Now let’s analyze the statements one by one:
(A) The motorcycle cost [tex]$12,000$[/tex] when purchased.
- When [tex]\( t = 0 \)[/tex], [tex]\( (0.75)^0 = 1 \)[/tex]. Substituting [tex]\( t = 0 \)[/tex] in the equation:
[tex]\[ V(0) = 12,000 \cdot 1 = 12,000 \][/tex]
- Therefore, the initial cost of the motorcycle was indeed [tex]$12,000$[/tex]. This statement is true.
(B) The motorcycle's value is decreasing at a rate of 0.25% each year.
- This can be checked by analyzing the base 0.75.
- A base of 0.75 indicates a 25% yearly decrease (since 0.75 means the motorcycle retains 75% of its value each year, hence it loses 25%).
- Because it decreases at a rate of 25% each year, not 0.25%, this statement is false.
(C) The motorcycle's value is decreasing at a rate of 75% each year.
- As previously discussed, 0.75 means the motorcycle retains 75% of its value each year.
- Thus, the motorcycle is losing 25% of its value each year, not 75%. This statement is false.
(D) The motorcycle cost [tex]$9000 when purchased.
- As shown under (A), the initial cost of the motorcycle was $[/tex]12,000. Thus, this statement is false.
Based on the analysis of all statements, the correct one is:
(A) The motorcycle cost $12,000 when purchased.
