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

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:

\[
V(y) = V_0 (1 + r)^y
\]

Where:
- $ V(y) $ is the value of the vehicle after $ y $ years.
- $ V_0 $ is the initial value (in this case, $75,500).
- $ r $ is the annual growth rate.
- $ y $ is the number of years after 2019.

### Step 2: Calculate the growth rate $ r $

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 $ r $:

\[
76,632.50 = 75,500(1 + r)
\]

Now, solve for $ r $:

\[
\frac{76,632.50}{75,500} = 1 + r
\]

\[
1.015 = 1 + r
\]

\[
r = 0.015
\]

### Step 3: Write the model

Using the growth rate $ r = 0.015 $, the model for the vehicle's value after $ y $ years is:

\[
V(y) = 75,500(1.015)^y
\]

### Matching to the options:
The correct model is:

\[
75500(1.015)^y
\]

This matches the third option: ***75500(1.015)^y***

Would you like further details or have any questions?

### 5 related questions:
1. How would the model change if the vehicle's value decreased exponentially?
2. How can you calculate the value of the vehicle after 5 years using this model?
3. What would be the vehicle's value in 2025 according to this model?
4. How does exponential growth differ from linear growth in terms of vehicle value?
5. 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!

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.

Learn how to model the exponential growth in value of a classic vehicle, appraised at $75,500 in 2019 and $76,632.50 in 2020. This problem uses exponential growth concepts to project the vehicle's future value and is suitable for students in Grades 10-12.

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