Problem 101
Question
The bar graph shows your chances of surviving to various ages once you reach 60 The functions $$ \begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned} $$ model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve Exercises \(101-102\) a. Find and interpret \(f(70)\) b. Find and interpret \(g(70)\) c. Which function serves as a better model for the chance of surviving to age \(70 ?\)
Step-by-Step Solution
Verified Answer
The values for \(f(70)\) and \(g(70)\) are 186% and 89% respectively. However, the better model for survival rate at the age of 70 is \(g(x)\) since it results in a plausible percentage within a 0 to 100% range.
1Step 1: Evaluate function f(x)
To find the value of \(f(70)\), substitute \(x=70\) in \(f(x) = -2.9x+286\). This results in \(f(70) = -2.9(70) + 286 = 186\)
2Step 2: Interpret Results for f(x)
The interpretation of the result of \(f(70) = 186\) means that, according to the function f, there is a 186% chance for a 60-year old surviving to age 70, which is incorrect as the survival rate cannot be more than 100%.
3Step 3: Evaluate function g(x)
To find the value of \(g(70)\), substitute \(x = 70\) in \(g(x) = 0.01x^2 - 4.9x + 370\). This results in \(g(70) = 0.01(70^2) - 4.9(70) + 370 = 89\)
4Step 4: Interpret Results for g(x)
The interpretation of the result of \(g(70) = 89\) means that, according to the function g, there is an 89% chance for a 60-year old surviving to age 70. This is a plausible percentage as it is within 0 to 100% range.
5Step 5: Comparing Models
Considering both results from f(70) and g(70), it is clear that g(x) provides a more plausible model for the chance of survival at the age of 70. The model f(x) resulted in a 186% survival rate, which is not possible.
Key Concepts
Function EvaluationInterpreting Mathematical ModelsReal-world Application of Algebra
Function Evaluation
Function evaluation is one of the fundamental skills in algebra, crucial for analyzing mathematical models. When we evaluate a function, we calculate the output for a specific input value. In the context of our survival model, evaluating the functions, represented by f(x) and g(x), gives us the probability of a person surviving to a particular age.
For example, the function f(x) = -2.9x + 286 tells us the chances of survival as a percentage. By substituting a specific age into the function, such as 70, we get f(70) = -2.9(70) + 286 = 186%. However, this result is not practical, as probabilities cannot exceed 100%. This shows us the importance of interpreting the results correctly to ensure they make sense within the context of the problem.
On the other hand, evaluating the function g(x) = 0.01x^2 - 4.9x + 370 at 70 gives us g(70) = 0.01(70)^2 - 4.9(70) + 370 = 89%, a more realistic outcome in terms of probability. It demonstrates that function evaluation doesn't just require calculation skills but also an understanding of the context to interpret the results accurately.
For example, the function f(x) = -2.9x + 286 tells us the chances of survival as a percentage. By substituting a specific age into the function, such as 70, we get f(70) = -2.9(70) + 286 = 186%. However, this result is not practical, as probabilities cannot exceed 100%. This shows us the importance of interpreting the results correctly to ensure they make sense within the context of the problem.
On the other hand, evaluating the function g(x) = 0.01x^2 - 4.9x + 370 at 70 gives us g(70) = 0.01(70)^2 - 4.9(70) + 370 = 89%, a more realistic outcome in terms of probability. It demonstrates that function evaluation doesn't just require calculation skills but also an understanding of the context to interpret the results accurately.
Interpreting Mathematical Models
Interpreting mathematical models is integral to applying algebra to real-world problems. Mathematical models use algebraic functions to represent relationships between different quantities. In our example regarding survival rates, we have two models, f(x) and g(x), that express the chance of surviving to a certain age.
Interpreting these models requires more than just mathematical computation; it involves understanding what the numbers and formulae mean in real life. The calculation of f(70) may lead to an incorrect interpretation without context, suggesting a survival rate above 100%, which is clearly not possible. In contrast, g(70) offers a more meaningful outcome, with an 89% chance of survival, which aligns with real-world expectations.
Moreover, interpreting models also means evaluating their validity. The model provided by function f(x) is invalidated by its output, indicating that not all mathematical models are suitable for every situation. This highlights the significance of critical thinking and validation when using algebra in practical applications.
Interpreting these models requires more than just mathematical computation; it involves understanding what the numbers and formulae mean in real life. The calculation of f(70) may lead to an incorrect interpretation without context, suggesting a survival rate above 100%, which is clearly not possible. In contrast, g(70) offers a more meaningful outcome, with an 89% chance of survival, which aligns with real-world expectations.
Moreover, interpreting models also means evaluating their validity. The model provided by function f(x) is invalidated by its output, indicating that not all mathematical models are suitable for every situation. This highlights the significance of critical thinking and validation when using algebra in practical applications.
Real-world Application of Algebra
Algebra not only solves theoretical problems but is invaluable in analyzing and making decisions based on real-world situations. By utilizing algebraic functions, we can model complex phenomena, such as survival rates among different age groups, and gain insights that inform policies, business, science, and personal decisions.
In our exercise, we have applied algebraic functions to determine the likelihood of a person's survival to age 70. The practicality of such models is seen in various fields, such as actuarial science, epidemiology, and even financial planning, where predicting life expectancy is essential.
However, real-world application demands precision and realism. As we saw in the example of f(x), not all models are created equal, and some can give improbable results. Therefore, while algebra is a powerful tool, it requires diligent application and interpretation to ensure that the outcomes are reasonable, valid, and useful for making informed decisions in real life.
In our exercise, we have applied algebraic functions to determine the likelihood of a person's survival to age 70. The practicality of such models is seen in various fields, such as actuarial science, epidemiology, and even financial planning, where predicting life expectancy is essential.
However, real-world application demands precision and realism. As we saw in the example of f(x), not all models are created equal, and some can give improbable results. Therefore, while algebra is a powerful tool, it requires diligent application and interpretation to ensure that the outcomes are reasonable, valid, and useful for making informed decisions in real life.
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