91Ó°ÊÓ

Growth in weight: The following table gives, for a certain man, his weight \(W=W(t)\) in pounds at age \(t\) in years. $$ \begin{array}{|c|c|} \hline t=\text { Age } \\ \text { (years) } & \begin{array}{c} W=\text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 4 & 36 \\ \hline 8 & 54 \\ \hline 12 & 81 \\ \hline 16 & 128 \\ \hline 20 & 156 \\ \hline 24 & 163 \\ \hline \end{array} $$ a. Make a table showing, for each of the 4-year periods, the average yearly rate of change in \(W\). b. Describe in general terms how the man's gain in weight varied over time. During which 4 -year period did the man gain the most in weight? c. Estimate how much the man weighed at age 30 . d. Use the average rate of change to estimate how much he weighed at birth. Is your answer reasonable?

Step by step solution

01

Find the Average Yearly Rate of Change

For each 4-year period, calculate the average rate of change in weight using the formula: \( \text{Average Rate of Change} = \frac{W_2 - W_1}{t_2 - t_1} \), where \(W_2\) and \(W_1\) are weights at ages \(t_2\) and \(t_1\) respectively. Apply this to each period (4-8, 8-12, 12-16, 16-20, 20-24).

02

Calculate for Each Period

Using the formula from Step 1:- For ages 4 to 8: \( \frac{54 - 36}{8 - 4} = 4.5 \text{ pounds/year}\).- For ages 8 to 12: \( \frac{81 - 54}{12 - 8} = 6.75 \text{ pounds/year}\).- For ages 12 to 16: \( \frac{128 - 81}{16 - 12} = 11.75 \text{ pounds/year}\).- For ages 16 to 20: \( \frac{156 - 128}{20 - 16} = 7 \text{ pounds/year}\).- For ages 20 to 24: \( \frac{163 - 156}{24 - 20} = 1.75 \text{ pounds/year}\).

03

Describe the Weight Gain Variations

The man's weight gain increased from age 4 to 16, with the highest average rate of change (11.75 pounds/year) between ages 12 and 16. However, after age 16, the rate of increase decreases to 7 pounds/year from 16 to 20, and then to 1.75 pounds/year from 20 to 24.

04

Estimate Weight at Age 30

Extrapolate using the average rate of change between ages 20 to 24 (1.75 pounds/year) beyond age 24. Projecting from age 24 to 30 involves 6 years:\( 163 + 6 \times 1.75 = 173.5 \) pounds at age 30.

05

Estimate Weight at Birth

Use the rate from the first period (4-8 years), assuming it held from 0 to 4 years for rough estimation.\( W_4 = 36 \) pounds at 4 years, so backward extrapolation from age 4:\( W_0 = 36 - 4 \times 4.5 = 18 \) pounds at birth.This is not a realistic birth weight, suggesting the rate varies more widely at early ages.

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Key Concepts

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

Weight Growth

Understanding weight growth over time is essential to grasp how an individual's body changes as they age. For the man in this exercise, weight growth is observed over several four-year intervals. These intervals reveal patterns in how quickly or slowly weight has been gained. During early childhood to young adulthood, the man’s weight increased steadily but at different rates. This natural variation in growth rates illustrates how different life stages can affect body changes and why it's essential to analyze each period separately. Keeping track of weight growth using these periods can help in examining trends and predicting future weight.

Extrapolation

Extrapolation is a mathematical technique used to estimate unknown values based on known data points. By projecting existing patterns beyond the observed data range, we can make informed predictions about future outcomes.
In the case of our exercise, to estimate the man's weight at age 30, we extrapolate using the average rate of weight change from the ages 20 to 24.

• This method assumes that the trend observed between ages 20 to 24 continues consistently into the future.
• While extrapolation is helpful, it relies heavily on the assumption that current patterns will persist. Sudden changes in daily habits, health, or growth spurts can affect accuracy.
• This emphasizes the importance of using extrapolation cautiously, especially when predicting biological or human factors.

Rate of Change Calculation

The rate of change calculation tells us how quickly something has increased or decreased over a specific time period. Particularly for weight, it provides insights into how fast a person is gaining or losing weight. To determine this rate, we use the formula: \( \text{Average Rate of Change} = \frac{W_2 - W_1}{t_2 - t_1} \) Where \(W_2\) and \(W_1\) are weights at different times \(t_2\) and \(t_1\). By plugging in actual values from our data table, we calculate the average rate of change of weight over different four-year spans:

• Example: From age 4 to 8, the calculation \(\frac{54 - 36}{8 - 4} = 4.5\) pounds/year shows the rate at which he put on weight for that period.

This calculation helps identify at which stages of life the man gained weight most rapidly and tailors insights accordingly.

Linear Approximation

Linear approximation is a technique used for estimating values of a function using a linear function to approximate its behavior over a short range.
In the context of our exercise, **linear approximation** helps us make predictions about the man's weight at unobserved ages by assuming a constant rate of change over small intervals. By examining the rates of change over four-year periods, we can approximate how much the man might weigh at any given time within those periods.

• The formula for the linear approximation in the exercise assumes linear behavior based on the segments provided.
• It further explains our logic for extrapolating and estimating the man's weight at age 30 or even retroactively guessing at birth weight, although the latter turned out to be unrealistic.

Such approximations are particularly useful when precise measurements are unavailable, allowing for practical decisions and future planning.