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Chapter 4: Problem 66

Asbestos and Cancer Insulation workers who were exposed to asbestos and employed before 1960 experienced an increased likelihood of lung cancer. If a group of insulation workers have a cumulative total of \(100,000\) years of work experience, with their first date of employment \(x\) years ago, then the number of lung cancer cases occurring within the group can be modeled by $$ N(x)=0.00437 x^{3.2} $$ (Source: Walker, A., Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视, Inc. \()\) (a) Calculate \(N(x)\) when \(x=5,10,\) and \(20 .\) What happens to the likelihood of cancer as \(x\) increases? (b) If \(x\) doubles, does the number of cancer cases also double?

Short Answer

Expert verified
N(x) increases significantly with x; doubling x more than doubles N(x).

Step by step solution

01

Calculate N(x) for x = 5

To find the number of lung cancer cases for insulation workers whose first employment was 5 years ago, substitute \(x = 5\) into the given model \(N(x) = 0.00437 x^{3.2}\). \[ N(5) = 0.00437 imes 5^{3.2} \] Calculate \(5^{3.2}\) first, and then multiply by 0.00437. \[ 5^{3.2} \approx 172.104 \] Now, calculate \[ N(5) = 0.00437 imes 172.104 \approx 0.751 \] Thus, \(N(5)\) is approximately 0.751.
02

Calculate N(x) for x = 10

To find the number of lung cancer cases for insulation workers whose first employment was 10 years ago, substitute \(x = 10\) into the model \(N(x) = 0.00437 x^{3.2}\).\[ N(10) = 0.00437 imes 10^{3.2} \] Calculate \(10^{3.2}\) first, and then multiply by 0.00437. \[ 10^{3.2} \approx 1584.893 \] Now, calculate \[ N(10) = 0.00437 imes 1584.893 \approx 6.922 \] Thus, \(N(10)\) is approximately 6.922.
03

Calculate N(x) for x = 20

To find the number of lung cancer cases for insulation workers whose first employment was 20 years ago, substitute \(x = 20\) into the model \(N(x) = 0.00437 x^{3.2}\).\[ N(20) = 0.00437 imes 20^{3.2} \] Calculate \(20^{3.2}\) first, and then multiply by 0.00437. \[ 20^{3.2} \approx 9050.860 \] Now, calculate \[ N(20) = 0.00437 imes 9050.860 \approx 39.555 \] Thus, \(N(20)\) is approximately 39.555.
04

Analyze the likelihood of cancer as x increases

Comparing \(N(5)\), \(N(10)\), and \(N(20)\), we see that they are approximately 0.751, 6.922, and 39.555 respectively. Notice how the number of cases significantly increases as \(x\) increases. This indicates that the likelihood of cancer increases more than proportionally with the number of years since the first employment because the model involves a power greater than 1 (3.2).
05

Check if doubling x doubles N(x)

Calculate \(N(2x)\) for \(x = 5\) and compare it with \(2 \times N(5)\). First, calculate \(N(10)\) previously found to be approximately 6.922, and \(2 \times N(5)\): \[ 2 \times N(5) = 2 \times 0.751 \approx 1.502 \]Since \(N(10)\) is approximately 6.922, which is far more than 1.502, doubling \(x\) does not simply double \(N(x)\). This is consistent for other values of \(x\) due to the exponent of 3.2 in the equation.

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Key Concepts

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

Exponential Growth
Exponential growth is a type of mathematical growth pattern where the rate of increase is proportional to the current value, leading to growth that accelerates over time.
It can be represented by functions of the form \( y = a \, b^x \), where \( a \) is the initial amount, \( b \) is the growth rate, and \( x \) is time.
In the context of the exercise, the model \( N(x) = 0.00437 x^{3.2} \), demonstrates exponential growth because as \( x \) increases, the power term \( x^{3.2} \) leads to a rapid rise in the number of lung cancer cases.
This is evident from the calculations where the values for \( N(x) \) increase from approximately 0.751 to 39.555 as \( x \) increases from 5 to 20.
  • Such growth happens because of the exponent 3.2, which is greater than 1.
  • Therefore, as time since first employment increases, the rate of new lung cancer cases accelerates, illustrating the principle of exponential growth.
  • This differs from linear growth, where the increase would be constant over time.
Understanding exponential growth helps explain why small increases in time lead to disproportionately larger increases in cancer risk in this model.
Cancer Risk Analysis
Cancer risk analysis involves assessing the probability of developing cancer over a period, often using statistical models like the one in this exercise.
The model \( N(x) = 0.00437 x^{3.2} \) helps to quantify the risk of lung cancer among asbestos-exposed workers based on the number of years since their first exposure.
This mathematical model gives a predictive insight into how long-term exposure affects cancer risk.
  • When \( x \) is small, the cancer risk as expressed by \( N(x) \) is lower.
  • As \( x \) increases, risk assessment shows a dramatic rise, indicating that longer durations of exposure lead to substantially higher cases of cancer.
  • The model is particularly insightful because it allows for prediction and analysis of cancer risk over time, guiding necessary health interventions for those at risk.
Additionally, doubling \( x \), or the time since exposure, does not equate to merely doubling the number of cases; this is evident from the insignificant scaling of a simple multiplicative factor due to the non-linearity expressed by the exponent.
Asbestos Exposure
Asbestos is a naturally occurring fibrous mineral once commonly used as insulation due to its resistance to heat.
However, it became known for its health hazards, particularly its link to lung cancer and other diseases.
When inhaled, asbestos fibers can become trapped in the lungs, leading to significant health risks over time.
  • Prolonged exposure to asbestos is strongly associated with increased cancer risk.
  • The exercise model underscores this by showing how lung cancer incidence increases as the years since first exposure rise.
  • The particles remain in the lungs for a long time, and the gradual cellular damage may eventually lead to cancer.
This prolonged latency period between exposure and disease manifestation makes asbestos particularly hazardous.
As a result, asbestos use is now highly regulated in most places, and awareness of its dangers is crucial in occupational health.

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