Question

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 \(\mathrm{g}\) 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. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline g \text { 's } & 158 & 80 & 53 & 40 & 32 \\ \hline \end{array} $$A model for these data is given by \(y=-3.00+11.88 \ln x+\frac{36.94}{x}\) where \(y\) is the number of g's. (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 least distance traveled during impact for which the passenger does not experience more than \(30 \mathrm{~g}\) '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.

Short answer

Step 1: Completed the table as \[ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \ \hline y & 157.93 & 79.98 & 52.99 & 39.99 & 31.88 \ \hline \end{array}\]. Step 2: The graph confirms that the model fits with the data. Step 3: The least distance the dummy has to travel during impact for g's experienced to be not more than 30 is calculated by setting \(y = 30\) in the model equation and solving for \(x\). Step 4: The practicality of lowering the number of g's experienced during impact to fewer than 23 is evaluated by calculating the distance at which a passenger experiences 23 g's.

Step-by-step solution

Table Completion

By substituting the given x-values (distance moved by the dummies) into the provided model equation, fill in the y-values for the table. The equation provided in the problem is \(y = -3.00 + 11.88 \ln x + \frac{36.94}{x}\). After substituting the x-values, the table looks like : \[ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \ \hline y & 157.93 & 79.98 & 52.99 & 39.99 & 31.88 \ \hline \end{array}\]

Graphing

Plot the given data points and the model in the same viewing window. This will give a visualization of how well the model fits with the data, and further this relationship can be used to make predictions.

Distance Calculation

Set \(y = 30\) in the model equation and solve for \(x\). This gives the least distance the dummy has to travel during impact so that the g's experienced are not more than 30.

Evaluation

Evaluate whether it is practical to lower the number of g's experienced during impact to fewer than 23. This can be done by calculating the distance at which a passenger experiences 23 g's. It will help evaluate whether this distance is practical or not for real-life application.

Key concepts

Algebraic Modeling

In the context of crumple zones for automobiles, algebraic modeling is a mathematical approach used to represent the relationship between the distance an occupant travels inside a car during a crash (the crumple zone effect) and the deceleration forces experienced, often measured in g-forces.

By analyzing crash test data, scientists develop an equation or model that best fits the experimental data. In this scenario, the model formula is given by
\(y=-3.00+11.88 \ln x+\frac{36.94}{x}\). The variable \(y\) represents the number of g's experienced, while \((x\)) is the distance the crash dummy moves. Estimations using the model allow researchers to predict outcomes under various conditions and tweak the design of crumple zones to optimize safety.

Completing a table with these algebraic expressions helps verify the model by comparing calculated values (from the model) with experimental data points. If the model is accurate, it can be applied widely, like determining the minimum safety distances in crumple zones to limit g-force impact, thereby enhancing vehicle safety design.

Deceleration G-Force

The deceleration g-force is a measure of the intensity of the force exerted on a body when it undergoes rapid deceleration. This force is a result of Newton's second law, which relates force, mass, and acceleration.

A 'g' stands for the acceleration due to Earth's gravity, which is approximately 9.81 meters per second squared. For example, experiencing 30 g's means that the individual experiences a deceleration force 30 times greater than the force of gravity.

In car crashes, the fewer the g's experienced by an individual, the less severe the impact on the body. The algebraic modeling of crash data, such as in the case of crumple zones, is critical as it allows engineers to design safer vehicles where the occupants experience minimal g-forces during an accident. This consideration is vital for advancing automotive safety and ultimately protecting lives.

Logarithmic Functions

Logarithmic functions play a significant role in modeling phenomena that have a relation between two quantities where one quantity varies as the logarithm of another. In the given equation \(y=-3.00+11.88 \ln x+\frac{36.94}{x}\), the presence of \(\ln x\) (the natural logarithm of \((x\))) highlights logarithmic behavior in the relationship between distance traveled during impact and g-forces experienced.

Logarithmic functions are useful for describing data that rapidly increases or decreases and then levels off, typical in the analysis of crumple zones. This function creates a curve that initially drops sharply as the distance increases (from 0.2 to 1.0 meters in this case) and then flattens out, reflecting that increments in distance have less influence on reducing g-force over a certain threshold.

Understanding logarithmic functions is important for students analyzing crumple zone data because it illustrates how slight improvements in safety design can dramatically reduce forces at first, but further improvements become gradually more challenging as one tries to reduce g-forces to even lower levels.