https://file.aiquickdraw.com/mathbot/file/1730820694463jg4h7u6u.jpg

To solve this problem, we need to find an exponential function of the form $ y = ab^x $ that best fits the given data and then use it to predict the debt at month $ x = 24 $.

The given data points are:
- $ (1, 620) $
- $ (4, 1083) $
- $ (5, 1215) $
- $ (7, 1902) $

### Step 1: Set Up Exponential Regression Model
Using an exponential regression model, we can use these points to determine the constants $ a $ and $ b $. To do this, we typically take the logarithm of both sides of the equation $ y = ab^x $ to transform it into a linear form. However, for simplicity and precision, let's use statistical tools to compute the exponential regression, which will provide accurate values for $ a $ and $ b $.

### Step 2: Calculate $ a $ and $ b $ Using Exponential Regression
After calculating using exponential regression, let's assume we find the constants $ a $ and $ b $. 

### Step 3: Predict Debt at Month $ x = 24 $
Once we have the values of $ a $ and $ b $, we substitute $ x = 24 $ into $ y = ab^x $ to find the predicted debt at month 24.

I'll perform these calculations to provide an accurate result.The predicted debt at month 24, using the exponential regression model, is approximately **$42,159**.

Would you like further details on how the calculations were performed, or do you have any questions?

Here are some related questions to explore:

1. How does an exponential model differ from a linear model in terms of growth prediction?
2. What are the effects of compound interest or exponential growth in real-world scenarios like credit card debt?
3. How can one calculate the rate of growth from the model's $ b $ parameter?
4. How accurate is exponential regression for small datasets, and what could improve its accuracy?
5. What other types of functions could model growth in debt over time?

**Tip:** Exponential growth is fast and often underestimated; always check growth assumptions, especially in financial contexts.

The table gives the amount of debt, in dollars, on an individual's credit card for certain months after opening the credit card. Using an exponential regression y = ab^x to model these data, what is the debt at month 24 predicted by the exponential function model, to the nearest dollar? (Assume that the debt continues and that no payments are made to reduce the debt.)

Learn how to use exponential regression to model credit card debt growth over time. This problem involves predicting debt at month 24 using the model y = ab^x based on initial debt values, assuming no payments are made. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation