The braking distance, \(y\) feet, for Damon's car to come to a complete stop is modeled by \(y=\frac{3\left(x^{2}+10 x\right)}{40},\) where \(x\) is the speed of the car in miles per hour. According to this model, which of the following is the maximum speed, in miles per hour, Damon can be driving so that the braking distance is less than or equal to 150 feet? \(\mathbf{F} .10\) \(\mathbf{G} .30\) \(\mathbf{H} .40\) \(\mathbf{J} .50\) \(\mathbf{K} .60\)

Grasping the concept of braking distance is crucial for safety in physics and real-world driving situations. The braking distance of a car is the distance it travels while coming to a complete stop after the brakes are applied. Mathematically, this can be represented as a function of the car's initial speed.

In our exercise, the formula is given by
\(y = \frac{3(x^2 + 10x)}{40}\),
where \(y\) stands for the braking distance in feet and \(x\) represents the car's speed in miles per hour (mph). This relation shows that the braking distance increases with the square of the speed - highlighting how even small increases in speed can significantly impact the stopping distance.

Students often find it easier to understand this concept by comparing different speeds and their corresponding braking distances. It is also beneficial to visualize the formula with a graph that plots speeds on the x-axis and braking distances on the y-axis, demonstrating the nonlinear relationship between the two variables.

Solving quadratic equations is a fundamental skill in algebra that applies to numerous real-life problems, including the analysis of braking distances. A quadratic equation is of the form
\(ax^2 + bx + c = 0\),
where the highest exponent of \(x\) is 2. To solve for \(x\) when the equation is set to equal zero involves factoring, completing the square, or using the quadratic formula:
\(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\).
In our case, we are dealing with a quadratic inequality, \(x^2 + 10x - 2000 \leq 0\), which required factoring to find the solutions that satisfy the inequality. When solving quadratic inequalities, it's important to visualize the solution(s) on a number line and identify the intervals where the inequality holds true, which is often the step that troubles students the most. For inequalities, remember to check the intervals between and outside the roots to determine where the solution lies.

Mathematical modeling plays an integral part in applying abstract mathematical concepts to real-world scenarios. It involves creating equations or functions that represent how systems or phenomena work. In our exercise, the braking distance formula is an example of mathematical modeling as it relates the speed of a car to the distance needed to stop.

Modeling helps in predicting behaviors and outcomes based on various conditions. In the context of our problem, Damon uses the model to understand safe driving practices. By solving the quadratic inequality derived from the braking distance formula, Damon can determine the highest safe speed at which the car can be driven.

Students learn best through modeling when they engage with tangible and concrete examples that connect the mathematical theory to everyday experiences. Incorporating interactive activities, such as simulations or experiments, can bolster this understanding. Modeling not only aids in problem-solving but also empowers students to apply mathematics in diverse fields, from physics to finance.