Math Problem Statement
A certain type of classic vehicle was appraised at $75,500 in 2019. In 2020, that same vehicle was valued at $76,632.50. Assuming an exponential growth in value, write a model to express the value of this particular vehicle after y years.
A certain type of classic vehicle was appraised at $75,500 in 2019. In 2020, that same vehicle was valued at $76,632.50. Assuming an exponential growth in value, write a model to express the value of this particular vehicle after y years.
0.015x + 75500
1132.50x + 75500
75500(1.015)y
75500(0.0148)y
Solution
We are given the value of a classic vehicle in 2019 and 2020, and asked to model its value assuming exponential growth.
Step 1: Recognize the exponential growth formula:
The general form for exponential growth is:
Where:
- is the value of the vehicle after years.
- is the initial value (in this case, $75,500).
- is the annual growth rate.
- is the number of years after 2019.
Step 2: Calculate the growth rate
From 2019 to 2020, the value of the car increased from $75,500 to $76,632.50. We can use the formula for exponential growth for a single year to solve for :
Now, solve for :
Step 3: Write the model
Using the growth rate , the model for the vehicle's value after years is:
Matching to the options:
The correct model is:
This matches the third option: 75500(1.015)^y
Would you like further details or have any questions?
5 related questions:
- How would the model change if the vehicle's value decreased exponentially?
- How can you calculate the value of the vehicle after 5 years using this model?
- What would be the vehicle's value in 2025 according to this model?
- How does exponential growth differ from linear growth in terms of vehicle value?
- How do we derive the growth rate from two known values in a different time period?
Tip:
When modeling exponential growth, small changes in the growth rate can lead to large differences in the future value over time!
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Algebra
Formulas
V(y) = V_0(1 + r)^y
Theorems
Exponential Growth Formula
Suitable Grade Level
Grades 10-12