Let's break down the problem and find the correct answers step-by-step:

Part (a): Finding the exponential model $y = a \cdot b^x$

Given the U.S. debt data, we need to model the debt as an exponential function of the form:

$y = a \cdot b^x$

where:

• $y$ is the debt in billions,
• $x$ is the number of years since 1900 (e.g., $x = 0$ in 1900, $x = 10$ in 1910, etc.),
• $a$ and $b$ are constants to be determined.

We can use two points from the data to estimate $a$ and $b$. Let's choose two points:

• $(x_1, y_1) = (0, 1)$ in 1900 (since $x = 0$ in 1900 and the debt was 1 billion dollars),
• $(x_2, y_2) = (109, 16,856)$ in 2013 (since $x = 109$ in 2013 and the debt was 16,856 billion dollars).

Using these two points, we can substitute them into the model equation $y = a \cdot b^x$.

1. For 1900 ($x = 0$, $y = 1$): $1 = a \cdot b^0 \Rightarrow a = 1$ So, the model becomes $y = b^x$.

2. For 2013 ($x = 109$, $y = 16,856$): $16,856 = 1 \cdot b^{109}$ Solving for $b$: $b^{109} = 16,856$ Taking the 109th root of both sides: $b = \sqrt[109]{16,856}$ Using a calculator, we get: $b \approx 1.0757$

Thus, the exponential model is: $y = 1.0757^x$

Part (b): Predict the debt in 2024

In 2024, $x = 124$ (since $x = 124$ in 2024 when starting from 1900). Using the model $y = 1.0757^x$, we can predict the debt:

$y = 1.0757^{124}$

Using a calculator, we get:

$y \approx 27,724.58 \, \text{billion dollars}$

Rounding to the nearest billion, the predicted debt in 2024 is:

$\boxed{27,725} \, \text{billion dollars}$

Part (c): Predict the year in which the debt will be $99 trillion ($99,000 billion)

We need to find the year $x$ when the debt will be 99,000 billion dollars. Using the model $y = 1.0757^x$, set $y = 99,000$ and solve for $x$:

$99,000 = 1.0757^x$

Take the natural logarithm of both sides:

$\ln(99,000) = x \ln(1.0757)$

Solving for $x$:

$x = \frac{\ln(99,000)}{\ln(1.0757)}$

Using a calculator:

$x \approx \frac{11.502}{0.07302} \approx 157.48$

Since $x = 0$ in 1900, the predicted year is:

$1900 + 157.48 \approx 2057$

Thus, the debt will reach $99 trillion around the year:

$\boxed{2057}$

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Let me know if you'd like further details or explanations!

Related Questions:

1. What is the general form of an exponential growth model?
2. How do you interpret the constants $a$ and $b$ in exponential models?
3. How do you solve an exponential equation using logarithms?
4. How would the debt growth model change if we included more complex factors like inflation?
5. Can this model be used indefinitely, or does it have limitations?

Tip:

Exponential models are often used to predict population growth, financial investments, and more, but they tend to overshoot predictions in the long term because they don't account for limiting factors. Always evaluate their validity over the range you're interested in.