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The function given is:

\[
M = 527 + 128.0\sqrt{t}, \quad 0 \leq t \leq 16
\]

where \(t\) represents the year, with \(t = 0\) corresponding to 1990.

### (a) Describe the transformation of the parent function \( f(x) = \sqrt{x} \):

- The base function is \( \sqrt{x} \).
- In the equation, the constant \( 128.0 \) scales the square root function vertically by a factor of 128.0.
- The addition of 527 translates the graph vertically upward by 527 units.

So, the correct description would include:
- **Vertical stretch of 128.0**
- **Vertical shift of 527 units upward**

### (b) Find the average rate of change of the function from 1990 to 2006.

The average rate of change is given by:

\[
\frac{M(t=16) - M(t=0)}{16 - 0}
\]

Let's compute the values:

For \( t = 16 \):
\[
M(16) = 527 + 128.0 \times \sqrt{16} = 527 + 128.0 \times 4 = 527 + 512 = 1039
\]

For \( t = 0 \):
\[
M(0) = 527 + 128.0 \times \sqrt{0} = 527 + 0 = 527
\]

Now, compute the average rate of change:
\[
\text{Average rate of change} = \frac{1039 - 527}{16} = \frac{512}{16} = 32
\]

Thus, the average rate of change is **32 billion miles per year**.

### (c) Rewrite the function so that \( t = 0 \) represents 2000.

Since \( t = 0 \) represents 1990, for 2000 we need to set \( t = 10 \) to correspond to 2000. So, the new function will be:

\[
M = 527 + 128.0\sqrt{t + 10}
\]

### (d) Use the model from part (c) to predict the number of miles driven by vans, pickups, and SUVs in 2024.

For 2024, \( t = 24 \) (since 2024 is 24 years after 1990).

We need to use the equation from part (c), with \( t = 24 - 10 = 14 \):

\[
M(14) = 527 + 128.0 \times \sqrt{14} 
\]

First, calculate \( \sqrt{14} \):

\[
\sqrt{14} \approx 3.74
\]

Now calculate \( M(14) \):

\[
M(14) = 527 + 128.0 \times 3.74 = 527 + 479.36 = 1006.36
\]

So, the predicted number of miles driven in 2024 is **1006.36 billion miles**.

---

Let me know if you would like more details or further clarification!

### 5 related questions:
1. What is the transformation if we have a horizontal shift in the function \( f(x) = \sqrt{x} \)?
2. How do you find the instantaneous rate of change for non-linear functions like \( \sqrt{x} \)?
3. How does a vertical stretch differ from a horizontal stretch in transformations?
4. What are the units for the average rate of change in real-world applications like this one?
5. Can we model other types of vehicle mileage trends using similar functions?

### Tip: Always start by identifying basic transformations such as shifts, stretches, and reflections when dealing with transformed parent functions like \( \sqrt{x} \).

Describe the transformation of the function M = 527 + 128.0√t, find the average rate of change from 1990 to 2006, rewrite the function to represent the year 2000, and predict the number of miles driven in 2024.

Learn how to analyze function transformations, calculate the average rate of change, rewrite functions to represent different years, and predict future values using mathematical models. This problem involves function transformations, rates of change, and the use of mathematical predictions for real-world scenarios.

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