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Chapter 2: Problem 33

CANCER SURVIVORS The number of living Americans who have had a cancer diagnosis has increased drastically since 1971 . In part, this is due to more testing for cancer and better treatment for some cancers. In part, it is because the population is older, and cancer is largely a disease of the elderly. The number of cancer survivors (in millions) between \(1975(t=0)\) and \(2000(t=25)\) is approximately $$ N(t)=0.0031 t^{2}+0.16 t+3.6 \quad(0 \leq t \leq 25) $$ a. How many living Americans had a cancer diagnosis in \(1975 ?\) In \(2000 ?\) b. Assuming the trend continued, how many cancer survivors were there in 2005 ?

Short Answer

Expert verified
a. In 1975, there were approximately \(3.6\) million living Americans who had a cancer diagnosis. In 2000, there were approximately \(10.8375\) million living Americans who had a cancer diagnosis. b. Assuming the trend continued, there were approximately \(15.66\) million cancer survivors in 2005.

Step by step solution

01

Calculate N(t) for t=0 (1975)

To find the number of cancer survivors in 1975, we need to find the value of N(t) when t=0. Plug t=0 into the given function and compute N(t): $$ N(0)=0.0031\times0^2 + 0.16\times0 + 3.6 $$
02

Calculate N(t) for t=25 (2000)

To find the number of cancer survivors in 2000, we need to compute the value of N(t) when t=25. Plug t=25 into the given function and calculate N(t): $$ N(25)=0.0031\times25^2 + 0.16\times25 + 3.6 $$ #b. Predicting the number of cancer survivors in 2005#
03

Calculate N(t) for t=30 (2005)

To predict the number of cancer survivors in 2005, we need to compute the value of N(t) when t=30, assuming the trend continues. Plug t=30 into the given function and calculate N(t): $$ N(30)=0.0031\times30^2 + 0.16\times30 + 3.6 $$

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Key Concepts

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

Understanding Cancer Statistics
Cancer statistics help us grasp the enormity of cancer's role in our lives. Such statistics often represent the number of people living with a cancer diagnosis, mortality rates, survivorship, and more. Several factors influence cancer statistics, including:
  • Increase in the elder population, as cancer is more common in older adults.
  • Advancements in medical testing that allow for earlier and more accurate diagnosis.
  • Improved treatments and follow-up care helping patients live longer post-diagnosis.
Understanding these statistical trends is essential in public health planning, supporting research efforts, and raising awareness about cancer.
Function Evaluation in Mathematical Modeling
When dealing with functions in mathematical modeling, function evaluation means finding the output by substituting specific input values. Finding the value of a function for a given input helps in understanding trends and making predictions. In our exercise, we have a mathematical model \( N(t) = 0.0031 t^2 + 0.16 t + 3.6 \), where \( N(t) \) indicates the number of cancer survivors in millions at time \( t \). To find the number of survivors for a certain year, simply plug the year (represented as \( t \)) into the function, and compute the value. This method makes it easier to evaluate how populations impacted by diseases like cancer change over time.
Interpreting Trends through Trend Analysis
Trend analysis in mathematical modeling involves analyzing the direction and pattern of data changes over time. This type of analysis helps predict future occurrences by observing historical data. Given data points, such as the number of cancer survivors over the years, we use trend analysis to determine if the numbers are increasing, decreasing, or stable. By examining the polynomial formula from the exercise, we see it's a quadratic function, indicating a potential non-linear increase in the number of cancer survivors, which makes sense considering improvements in healthcare. Understanding this growth pattern aids in better planning for healthcare provision and resources.
Predictive Mathematics in Health Trends
Predictive mathematics utilizes mathematical formulations to foresee future events or metrics based on current and historical data. By applying predictive models like the quadratic function in our case, we can make reasonable assumptions about future cancer survivorship. For instance, using the model provided, predicting the number of survivors in 2005 involves assuming current trends will continue. This is crucial for healthcare planning, as understanding potential future values of cancer statistics enables policymakers to implement strategies that address possible increases in healthcare needs. In conclusion, predictive mathematics can provide valuable estimates that inform proactive health policies.

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