Question

The value of a motorcycle changes according to the equation: V (t) = 5,000(1.03)^t , where V (t) = value in dollars and t = time in years.
a) In the equation, the number 5,000 represent the ___ of the motorcycle.
b) What does V(t) = 9,000 mean in the context of this problem?
c) What does a blue of t = 3 mean in the context of this problem?
Answer
The options for part b were: The value of the motorcycle is 9000, the initial value of the motercycle is 9000, the time it takes for the motercycles value to be 9000, the rate at which it takes the motercycle to be worth 9000
Answered by GPT-4o mini
a) In the equation \( V(t) = 5,000(1.03)^t \), the number 5,000 represents the **initial value** of the motorcycle when \( t = 0 \). This is the value of the motorcycle at the beginning of the time period being considered (i.e., when it is first purchased).

b) The equation \( V(t) = 9,000 \) means that the value of the motorcycle reaches **$9,000 at a certain time \( t \)**. This indicates that the motorcycle's value has increased to $9,000 after a certain number of years have passed.

c) A value of \( t = 3 \) in the context of this problem means that **3 years** have passed since the initial purchase of the motorcycle. Specifically, it can be used to find the value of the motorcycle after 3 years by substituting \( t = 3 \) into the equation \( V(t) = 5,000(1.03)^t \).
Answered by GPT-4o mini
Given the options you provided for part b, the correct interpretation of \( V(t) = 9,000 \) would be:

**The value of the motorcycle is 9000.**

This statement clearly indicates that the motorcycle's value has reached $9,000 at some specific time \( t \), aligning with the understanding that \( V(t) \) denotes the value of the motorcycle at time \( t \).