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Modeling the path of a roller coaster using a polynomial function is an exciting way to visualize the fun and thrilling world of amusement park rides. In essence, each term in the polynomial function represents a different aspect of the track's shape and how it changes as you move along the x-axis. The given polynomial for our problem is: \[ f(x) = 0.001x^3 - 0.12x^2 + 3.6x + 10 \]

• The term \(0.001x^3\) affects the curvature, making the track bend upwards or downwards.
• The term \(-0.12x^2\) contributes to the steepness and direction, moving the track in a different curvature compared to the cubic term.
• The linear term \(3.6x\) indicates a slope, which directly influences the incline or decline of the track.
• The constant \(10\) adjusts the starting height of the roller coaster.

These terms combine to create a model that suggests dips and rises, much like the actual movements experienced on a roller coaster. This powerful mathematical representation helps engineers design thrilling and safe rides. By plugging in different x-values, you can see how high the track is at specific points, thus predicting the ride's experience.

The substitution method is a straightforward technique utilized for calculating the value of a function at a specific point. You essentially replace the variable in the function with a given number. This is precisely what was applied in the task for different values of \(x\).For instance, to find the height of the track when \(x=20\):1. Substitute \(x=20\) into the polynomial: \[ f(20) = 0.001(20)^3 - 0.12(20)^2 + 3.6(20) + 10 \] 2. Calculate each component step-by-step: - The cube of 20 is multiplied by 0.001, resulting in 8. - The square of 20 is multiplied by -0.12, giving -48. - And 20 times 3.6 equals 72.3. Add these results together along with the constant: \[ 8 - 48 + 72 + 10 = 42 \]Using the substitution method is handy because it simplifies one component at a time, reducing the risks of mistakes. This approach is vital for students learning to handle not only simple problems but also complex polynomials in both academics and real-life applications, like roller coaster design.

Evaluating a polynomial function involves finding the result of the polynomial at specific input values, which was demonstrated in the original solution through various x-values. To grasp the concept, consider evaluating the polynomial:\[ f(x) = 0.001x^3 - 0.12x^2 + 3.6x + 10 \]To evaluate it at \(x=40\), break it down:

• Calculate the cube of 40 and multiply it by 0.001, resulting in 64.
• Calculate the square of 40 and multiply it by -0.12, resulting in -192.
• Multiply 40 by 3.6 to get 144.
• Add those terms to the constant 10.

All combined, this gives the evaluated result:\[ f(40) = 64 - 192 + 144 + 10 = 26 \]This process requires careful attention to detail, ensuring each term is correctly evaluated and combined. Polynomial evaluation is a key skill in mathematics because it lets us predict and analyze real-world phenomena, from engineering projects like roller coasters to complex natural occurrences.