 Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. 
 A cyclist is riding on a path whose elevation is modeled by the function 
f(x)=0.08(16x−x 
2
 )
where x and are measured in miles. Find the rate of change of elevation when x = 4. 

To determine the rate of change of elevation at \( x = 4 \), we need to find the derivative of the given elevation function \( f(x) \). Calculus is required here because the "rate of change" refers to the derivative, which measures how the function's output (elevation) changes with respect to its input (distance).

The given function is:
\[
f(x) = 0.08(16x - x^2)
\]

### Step 1: Differentiate the function
We use the power rule to differentiate the function with respect to \( x \).

First, expand the expression:
\[
f(x) = 0.08(16x - x^2) = 1.28x - 0.08x^2
\]

Now differentiate term by term:
\[
f'(x) = \frac{d}{dx}(1.28x) - \frac{d}{dx}(0.08x^2)
\]

\[
f'(x) = 1.28 - 0.16x
\]

### Step 2: Evaluate the derivative at \( x = 4 \)
Substitute \( x = 4 \) into the derivative:
\[
f'(4) = 1.28 - 0.16(4)
\]
\[
f'(4) = 1.28 - 0.64
\]
\[
f'(4) = 0.64
\]

### Conclusion:
The rate of change of elevation when \( x = 4 \) miles is \( 0.64 \) miles per mile (which can also be interpreted as a slope of 0.64).

#### Would you like further details or have any questions?

Here are some related questions:
1. What is the physical interpretation of a derivative in this context?
2. How does the rate of change vary with respect to distance for this function?
3. What happens to the elevation function as \( x \) increases beyond 4 miles?
4. Can we find where the maximum elevation occurs using calculus?
5. How would you approximate the derivative numerically if you did not know calculus?

**Tip**: The derivative of a function at a point gives the slope of the tangent line to the graph at that point, which is the rate of change of the function at that specific value.

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. A cyclist is riding on a path whose elevation is modeled by the function f(x) = 0.08(16x - x^2) where x and f(x) are measured in miles. Find the rate of change of elevation when x = 4.

Learn how to find the rate of change of elevation for a cyclist's path modeled by the function f(x) = 0.08(16x - x^2). This calculus problem demonstrates the use of derivatives to calculate the slope at a specific point, x = 4, providing a step-by-step solution. Suitable for advanced high school calculus students (Grades 11-12).

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