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Chapter 1: Problem 11

Growth in height: The following table gives, for a certain man, his height \(H=H(t)\) in inches at age \(t\) in years. $$ \begin{array}{|c|c|} \hline \begin{array}{c} t=\text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} H=\text { Height } \\ \text { (inches) } \end{array} \\ \hline 0 & 21.5 \\ \hline 5 & 42.5 \\ \hline 10 & 55.0 \\ \hline 15 & 67.0 \\ \hline 20 & 73.5 \\ \hline 25 & 74.0 \\ \hline \end{array} $$ a. Use functional notation to express the height of the man at age 13 , and then estimate its value. b. Now we study the man's growth rate. i. Make a table showing, for each of the 5-year periods, the average yearly growth ratethat is, the average yearly rate of change in \(H\). ii. During which 5 -year period did the man grow the most in height? iii. Describe the general trend in the man's growth rate. c. What limiting value would you estimate for the height of this man? Explain your reasoning in physical terms.

Short Answer

Expert verified
a. \( H(13) \approx 62.2 \) inches. b.i. Maximum growth 0-5 years, ii. Growth decreased over time. c. Limiting height ~74 inches.

Step by step solution

01

Expressing Height at Age 13 Using Functional Notation

To find the height of the man at age 13 using functional notation, we denote it as \( H(13) \). Since the height function \( H(t) \) is not explicitly given, we can estimate \( H(13) \) by assuming linear interpolation between known values (ages 10 and 15):\[ H(13) \approx H(10) + \frac{13 - 10}{15 - 10}(H(15) - H(10)) \]Substitute the given heights: \[ H(13) \approx 55 + \frac{3}{5} imes (67 - 55) = 55 + \frac{3}{5} imes 12 = 55 + 7.2 = 62.2 \]Thus, \( H(13) \approx 62.2 \) inches.
02

Calculating Average Yearly Growth Rates

We calculate the average yearly growth rate for each 5-year interval using the formula: \[ \text{Average Growth Rate} = \frac{H(t_2) - H(t_1)}{t_2 - t_1} \]- From age 0 to 5: \[ \frac{42.5 - 21.5}{5 - 0} = \frac{21}{5} = 4.2 \text{ inches/year} \]- From age 5 to 10: \[ \frac{55 - 42.5}{10 - 5} = \frac{12.5}{5} = 2.5 \text{ inches/year} \]- From age 10 to 15: \[ \frac{67 - 55}{15 - 10} = \frac{12}{5} = 2.4 \text{ inches/year} \]- From age 15 to 20: \[ \frac{73.5 - 67}{20 - 15} = \frac{6.5}{5} = 1.3 \text{ inches/year} \]- From age 20 to 25: \[ \frac{74 - 73.5}{25 - 20} = \frac{0.5}{5} = 0.1 \text{ inches/year} \]
03

Finding Period of Maximum Growth

By comparing the average yearly growth rates calculated, we observe that the period from age 0 to 5 years has the highest growth rate of 4.2 inches per year. Thus, the man grew the most during this interval.
04

General Trend in Growth Rate

The data shows a decreasing trend in the average yearly growth rate as the man ages: from 4.2 inches per year at age 0-5 to just 0.1 inches per year at age 20-25. This indicates that as the man grows older, his growth rate slows down markedly.
05

Estimating Limiting Value of Height

Based on the trend observed, the man's height appears to stabilize around 74 inches at age 25, with a minuscule growth rate hinting that he is nearing his maximum height. Physically, this suggests that the man likely reaches his full adult size at this age, with the height approaching a limiting value of approximately 74 inches.

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Key Concepts

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

Linear Interpolation
In the exercise, linear interpolation is used to estimate the height of a man at age 13, based on known height values at ages 10 and 15. Linear interpolation is a method to estimate the value of a function between two known values. It assumes that the change between two known values is linear, or in a straight line.

What this really means is that if you know the height at age 10 and at age 15, you can make an educated guess about the height at age 13 by drawing a straight line between the two known points on the graph and choosing the value directly under age 13.
  • Imagine you have two points in a straight line: (10, 55) and (15, 67).
  • To find the point at age 13, plug into the formula: \[ H(13) \approx H(10) + \frac{13 - 10}{15 - 10}(H(15) - H(10)) \]
  • This calculation gives you an estimate of 62.2 inches.
Remember, linear interpolation is a basic form of estimation and works well when changes are smooth between data points.
Average Yearly Growth Rate
The average yearly growth rate provides an idea of how much height is gained on average each year over a certain period. This exercise uses the average growth rate to analyze the man's growth over various 5-year intervals.

The calculation involves taking the difference in heights at the beginning and end of a period and dividing by the number of years in that period.
  • Example: From age 0 to 5, the calculation is: \[ \frac{42.5 - 21.5}{5} = 4.2 \text{ inches per year} \]
  • Each five-year interval shows different growth rates.
  • The highest growth rate occurs from age 0 to 5 at 4.2 inches per year, highlighting a period of rapid growth.
Growth rates typically decrease as a person gets older, reflecting how growth slows with age. Calculating average growth helps in understanding and predicting future growth patterns.
Rate of Change
The rate of change in this context refers to how quickly height changes over time. It provides insight into the dynamic nature of growth by supplying a numerical value to describe this change.

In appreciating the rate of change, we see that different stages of life present different growth rates. Initially, there's rapid growth; then, as time passes, this slows significantly. The essence here is captured through the average growth rates calculated in the exercise.
  • It depicts how 4.2 inches per year from ages 0 to 5 dwindles down to just 0.1 inches per year from ages 20 to 25.
  • This slowing down signifies the natural biological limit as individuals reach maturity.
Understanding rate of change is crucial as it helps explain trends over time and sets expectations for growth-related milestones in human development.

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Most popular questions from this chapter

How much can I borrow? The function in Example \(1.2\) can be rearranged to show the amount of money \(P=P(M, r, t)\), in dollars, that you can afford to borrow at a monthly interest rate of \(r\) (as a decimal) if you are able to make \(t\) monthly payments of \(M\) dollars: $$ P=M \times \frac{1}{r} \times\left(1-\frac{1}{(1+r)^{r}}\right) . $$ Suppose you can afford to pay \(\$ 350\) per month for 4 years. a. How much money can you afford to borrow for the purchase of a car if the prevailing monthly interest rate is \(0.75 \%\) ? (That is \(9 \%\) APR.) Express the answer in functional notation, and then calculate it. b. Suppose your car dealer can arrange a special monthly interest rate of \(0.25 \%\) (or \(3 \%\) APR). How much can you afford to borrow now? c. Even at \(3 \%\) APR you find yourself looking at a car you can't afford, and you consider extending the period during which you are willing to make payments to 5 years. How much can you afford to borrow under these conditions?

The American food dollar: The following table shows the percentage \(P=P(d)\) of the American food dollar that was spent on eating away from home (at restaurants, for example) as a function of the date \(d\). $$ \begin{array}{|c|c|} \hline d=\text { Year } & \begin{array}{c} P=\text { Percent spent } \\ \text { away from home } \end{array} \\ \hline 1960 & 19 \% \\ \hline 1980 & 27 \% \\ \hline 2000 & 37 \% \\ \hline \end{array} $$ a. Find \(P(1980)\) and explain what it means. b. What does \(P(1990)\) mean? Estimate its value. c. What is the average rate of change per year in percentage of the food dollar spent away from home for the period from 1980 to 2000 ? d. What does \(P(1997)\) mean? Estimate its value. (Hint: Your calculation in part \(c\) should be useful.)e. Estimate the value of \(P(2003)\) and explain how you made your estimate.

United States population growth: In 1960 the population of the United States was about \(180 \mathrm{mil-}\) lion. Since that time the population has increased by approximately \(1.2 \%\) each year. This is a verbal description of the function \(N=N(t)\), where \(N\) is the population, in millions, and \(t\) is the number of years since 1960 . a. Express in functional notation the population of the United States in 1963. Calculate its value. b. Use the verbal description of \(N\) to make a table of values that shows U.S. population in millions from 1960 through \(1965 .\) c. Make a graph of U.S. population versus time. Be sure to label your graph appropriately. d. Verify that the formula \(180 \times 1.012^{t}\) million people, where \(t\) is the number of years since 1960 , gives the same values as those you found in the table in part b. (Note: Because \(t\) is the number of years since 1960, you would use \(t=2\) to get the population in 1962.) e. Assuming that the population has been growing at the same percentage rate since 1960 , what value does the formula above give for the population in 2000? (Note: The actual population in 2000 was about 281 million.)

Production rate: The total number \(t\) of items that a manufacturing company can produce is directly proportional to the number \(n\) of employees. a. Choose a letter to denote the constant of proportionality, and write an equation that shows the proportionality relation. b. What in practical terms does the constant of proportionality represent in this case?

Tax owed: The income tax \(T\) owed in a certain state is a function of the taxable income \(I\), both measured in dollars. The formula is \(T=0.11 I-500\) a. Express using functional notation the tax owed on a taxable income of \(\$ 13,000\), and then calculate that value. b. If your taxable income increases from \(\$ 13,000\) to \(\$ 14,000\), by how much does your tax increase? c. If your taxable income increases from \(\$ 14,000\) to \(\$ 15,000\), by how much does your tax increase?
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