4. I bought $10 worth of pokémon cards in 2000 and in 2022 I found out that they are now worth $180. (a) Assuming they appreciate value exponentially, create an model p(t) = poert that repre- sents this situation, by solving for the constants.
4. I bought $10 worth of pokémon cards in 2000 and in 2022 I found out that they are now worth $180. (a) Assuming they appreciate value exponentially, create an model p(t) = poert that repre- sents this situation, by solving for the constants.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Appreciating Value of Pokémon Cards: An Exponential Model
#### Problem Statement:
**4.** I bought $10 worth of Pokémon cards in 2000 and in 2022 I found out that they are now worth $180.
**(a)** Assuming they appreciate in value exponentially, create a model \( p(t) = p_0 e^{rt} \) that represents this situation by solving for the constants.
#### Solution:
To create an exponential model \( p(t) = p_0 e^{rt} \), follow these steps to solve for the constants \( p_0 \) and \( r \):
1. Identify the initial value \( p_0 \):
- \( p_0 \) is the initial value you paid for the Pokémon cards in 2000, which is $10.
2. Use the given information about the value in 2022:
- The value in 2022 is $180.
- The time elapsed \( t \) from 2000 to 2022 is \( 2022 - 2000 = 22 \) years.
3. Set up the equation with these values:
\[
180 = 10 \cdot e^{22r}
\]
4. Solve for the rate \( r \):
\[
\frac{180}{10} = e^{22r}
\]
\[
18 = e^{22r}
\]
\[
\ln(18) = 22r
\]
\[
r = \frac{\ln(18)}{22}
\]
So, the value appreciation rate \( r \) can be computed as:
\[
r \approx \frac{2.890}{22} \approx 0.13136
\]
#### Final Model:
Thus, the exponential model representing the value appreciation of the Pokémon cards is:
\[
p(t) = 10 \cdot e^{0.13136t}
\]
You can use this model to predict the value of the Pokémon cards at any future point in time \( t \) years after 2000.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7618841-dd70-4dc8-afce-d1557e29b2d5%2F8b48a7ab-40b8-4b5a-bd94-56a14099a10a%2Fq6lui6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Appreciating Value of Pokémon Cards: An Exponential Model
#### Problem Statement:
**4.** I bought $10 worth of Pokémon cards in 2000 and in 2022 I found out that they are now worth $180.
**(a)** Assuming they appreciate in value exponentially, create a model \( p(t) = p_0 e^{rt} \) that represents this situation by solving for the constants.
#### Solution:
To create an exponential model \( p(t) = p_0 e^{rt} \), follow these steps to solve for the constants \( p_0 \) and \( r \):
1. Identify the initial value \( p_0 \):
- \( p_0 \) is the initial value you paid for the Pokémon cards in 2000, which is $10.
2. Use the given information about the value in 2022:
- The value in 2022 is $180.
- The time elapsed \( t \) from 2000 to 2022 is \( 2022 - 2000 = 22 \) years.
3. Set up the equation with these values:
\[
180 = 10 \cdot e^{22r}
\]
4. Solve for the rate \( r \):
\[
\frac{180}{10} = e^{22r}
\]
\[
18 = e^{22r}
\]
\[
\ln(18) = 22r
\]
\[
r = \frac{\ln(18)}{22}
\]
So, the value appreciation rate \( r \) can be computed as:
\[
r \approx \frac{2.890}{22} \approx 0.13136
\]
#### Final Model:
Thus, the exponential model representing the value appreciation of the Pokémon cards is:
\[
p(t) = 10 \cdot e^{0.13136t}
\]
You can use this model to predict the value of the Pokémon cards at any future point in time \( t \) years after 2000.
Transcribed Image Text:### Educational Website Content
**Task Instructions:**
1. **Part B:**
- **Objective:** Use the provided model to determine the time period required for the value to reach $1000.
- **Instructions:** Apply the given mathematical or statistical model to calculate after how many years the target value of $1000 will be achieved. Ensure to follow the model's steps accurately and show your work for verification.
2. **Part C:**
- **Scenario:** The Pokémon card collection is planned to be left as an inheritance.
- **Objective:** Estimate the future value of this Pokémon card collection in the year 2100.
- **Instructions:** Using the model, project the value of the card collection to the year 2100. Analyze the growth trend and economic assumptions used in the model to reach a justified estimation.
**Note:** This exercise requires understanding and applying future value models, considering factors like interest rates, inflation, or other growth metrics relevant to the valuation of collectibles.
---
**Graphical/Diagram Explanation (if applicable):**
- N/A
**End of Content**
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