https://file.aiquickdraw.com/mathbot/file/1727770072861c65z4dzd.jpg

We are given the polynomial model:

\[
y = -0.0001(x^2 - 100x)
\]

where \( 0 \leq x \leq 100 \), and \( y \) is the height reached by the roller coaster.

### Step 1: Expand the polynomial model
First, expand the expression inside the parentheses:

\[
y = -0.0001(x^2 - 100x) = -0.0001x^2 + 0.01x
\]

So, the model becomes:

\[
y = -0.0001x^2 + 0.01x
\]

### Step 2: Find the maximum height
To find the highest point reached by the roller coaster, we need to find the maximum value of this quadratic function. Since the coefficient of \(x^2\) is negative (\(-0.0001\)), the parabola opens downwards, meaning it has a maximum point.

To find the \(x\)-coordinate of the maximum, use the formula for the vertex of a parabola:

\[
x_{\text{max}} = \frac{-b}{2a}
\]

where \(a = -0.0001\) and \(b = 0.01\).

\[
x_{\text{max}} = \frac{-0.01}{2(-0.0001)} = \frac{-0.01}{-0.0002} = 50
\]

So, the maximum height is reached when \(x = 50\).

### Step 3: Calculate the maximum height
Substitute \(x = 50\) into the equation to find the maximum height:

\[
y = -0.0001(50^2) + 0.01(50)
\]
\[
y = -0.0001(2500) + 0.01(50) = -0.25 + 0.5 = 0.25
\]

Thus, the highest point reached by the roller coaster is **0.25 units**.

Would you like more details or have any other questions?

Here are some related questions:

1. How does the coefficient of \(x^2\) affect the shape of a parabola?
2. Can this polynomial model be used for all roller coaster designs?
3. What would happen if the coefficient of \(x^2\) was positive?
4. How do you determine the domain of a polynomial model like this?
5. What other types of models can be used for roller coaster height predictions?

**Tip:** The vertex formula for a parabola is a quick way to find the maximum or minimum value without fully solving the quadratic equation.

A designer of roller coasters decided to use a polynomial model for a new construction and tried the polynomial model y = -0.0001(x^2 - 100x), where 0 ≤ x ≤ 100 and y is the height reached. What is the highest point reached by the roller coaster?

This problem uses a quadratic polynomial model to find the highest point reached by a roller coaster. We apply the vertex formula to determine the maximum height. Suitable for high school students learning about quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation