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Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g's the crash victims experience. (One \(g\) is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g's.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g's experienced during deceleration by crash dummies that were permitted to move \(x\) meters during impact. The data are shown in the table. A model for the data is given by \(y=-3.00+11.88 \ln x+(36.94 / x),\) where \(y\) is the number of g's. $$ \begin{array}{|c|c|} \hline x & \text { g's } \\ \hline 0.2 & 158 \\ 0.4 & 80 \\ 0.6 & 53 \\ 0.8 & 40 \\ 1.0 & 32 \\ \hline \end{array} $$ (a) Complete the table using the model. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline y & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed \(30 \mathrm{~g}\) 's. (d) Do you think it is practical to lower the number of g's experienced during impact to fewer than \(23 ?\) Explain your reasoning.

Step by step solution

01

Calculate \(y\) for Given \(x\) Values

For each value of \(x\) from the table, we substitute into the given formula \(y = -3.00+11.88 \ln x+(36.94 / x)\) to calculate the corresponding \(y\) value which represents the number of g's.

02

Compare Graphical Representation

Plot the data points and the model on a graph using a graphing utility. This allows us to visually compare the accuracy of the model with actual data points.

03

Estimate Distance for 30 g's

Set the equation equal to 30 and solve for \(x\) (distance traveled during impact). This gives an estimate of the distance required to keep passenger deceleration under 30 g's.

04

Practicality Analysis

Discuss the practicality of reducing the number of g's experienced during impact to fewer than 23. This involves understanding the physical implications and limitations in the context of a car crash.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Crumple Zones

Crumple zones are an essential safety feature in the design of modern automobiles. They are areas of a vehicle that are designed to deform and crumple in a controlled way during a collision. The main purpose of crumple zones is to absorb some of the energy of the impact, preventing it from being transmitted to the occupants of the vehicle.

Crumple zones work by extending the time over which the car comes to a stop during a crash. This reduced deceleration results in lower g-forces acting on the passengers, which in turn decreases the likelihood of serious injury. When crumple zones are properly designed, they can make a significant difference in the survivability of high-speed impacts.

G-Force

The term 'g-force' refers to the measurement of acceleration felt as weight. It is a unit of force that indicates the amount of pressure exerted on an object when it accelerates, relative to the force of gravity. One g is equivalent to the acceleration due to gravity at the Earth's surface, approximately 9.81 meters per second squared.

In the context of car crashes, g-force levels indicate the intensity of the forces acting on passengers during impact. High g-forces can result in serious injuries or fatalities, which is why managing deceleration through features like crumple zones is critical for passenger safety. Humans can withstand high g-forces for very short durations, but reducing the g-force experienced during a car crash is essential for protecting occupants.

Impact Deceleration Calculation

Impact deceleration calculation involves determining the rate of deceleration a passenger experiences during a car crash. This is crucial for assessing the efficacy of safety features and for designing vehicles that minimize the risk of injury. The g-force experienced by crash dummies or passengers can be calculated using data that measures the distance traveled during impact.

The mathematical model provided in the problem, represented by the equation
\[y = -3.00 + 11.88 \ln x + (36.94 / x),\]
is used to estimate the g-force (y) based on the distance traveled (x). By inputting various distances into this formula, manufacturers and safety researchers can predict how many g's occupants will experience, and thus design better safety measures accordingly.

Mathematical Modeling in Physics

Mathematical modeling is a fundamental tool in physics that allows scientists and engineers to create abstract representations of real-world processes. These models are composed of mathematical symbols and equations that describe relationships among various physical quantities.

In the case of car crashes, mathematical modeling helps in predicting the outcomes of different crash scenarios. The model
\[y = -3.00 + 11.88 \ln x + (36.94 / x)\]
is an example of how logarithmic and rational functions can be combined to approximate complex physical phenomena such as the deceleration forces experienced by car crash dummies. These models are essential for developing safer vehicles and can be critical in setting industry safety standards.

Graphing Utility Comparison

Graphing utilities are powerful tools that help in the visual representation of mathematical models and data. They allow one to plot equations and compare them with actual data points to assess the validity of the model. Graphing the model \(y = -3.00 + 11.88 \ln x + (36.94 / x)\) and the empirical data allows us to see how well the model predicts the g-forces experienced at different distances traveled during a car crash.

By comparing the graphical output of the actual data and the model, we can gauge the model's accuracy. A close match between the data points and the curve suggests that the model is a good representation of reality. This graphical comparison is often an iterative step in the mathematical modeling process, leading to refinements and better predictions of physical events like car crashes.