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Chapter 1: Problem 4

Swimming records: The world record time for a certain swimming event was \(63.2\) seconds in 1950 . Each year thereafter, the world record time decreased by \(0.4\) second. a. Use a formula to express the world record time as a function of the time since 1950 . Be sure to explain the meaning of the letters you choose and the units. b. Express using functional notation the world record time in the year 1955 , and then calculate that value. c. Would you expect the formula to be valid indefinitely? Be sure to explain your answer.

Short Answer

Expert verified
a) \( R(t) = 63.2 - 0.4t \). b) \( R(5) = 61.2 \) seconds. c) The formula is not valid indefinitely due to physical limits.

Step by step solution

01

Define the Variables

Let \( t \) be the number of years since 1950, which means \( t = 0 \) corresponds to the year 1950. Let \( R(t) \) represent the world record time as a function of \( t \). The initial record time is \( 63.2 \) seconds.
02

Construct the Function Formula

The world record decreases by \( 0.4 \) seconds each year. Therefore, the function \( R(t) \) is given by \( R(t) = 63.2 - 0.4t \). This formula represents a linear relationship where every increase in \( t \) by one year results in a decrease of \( 0.4 \) seconds in record time.
03

Calculate World Record Time for 1955

The year 1955 corresponds to \( t = 5 \) years after 1950. Substitute \( t = 5 \) into the function: \[ R(5) = 63.2 - 0.4 \times 5 = 63.2 - 2.0 = 61.2 \] So, the world record time in 1955 was \( 61.2 \) seconds.
04

Evaluate the Validity of the Formula

The formula \( R(t) = 63.2 - 0.4t \) assumes a constant rate of decrease and does not consider physiological or technical limits of human performance. Over many years, physical limits mean that records cannot continue to decrease indefinitely by a fixed amount. Thus, the formula may not be valid indefinitely.

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Key Concepts

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

Function Notation
Function notation is a convenient way to express a mathematical relationship. In this exercise, we are dealing with the world record time for a swimming event. We want to express this as a function of time since 1950.

We define a variable \( t \) that represents the number of years since 1950. So, \( t = 0 \) when it is 1950. The function representing the world record time can be denoted as \( R(t) \). Here,
  • \( R(t) \) stands for the world record time in seconds.
  • \( t \) is the number of years after 1950.
Using function notation, we can easily plug in different values of \( t \) to find the record for different years. It simplifies the process of calculation and communication of the relationship.
Rate of Change
The rate of change in this context refers to how much the world record is decreasing each year. In the problem, it's specified that the record time decreases by 0.4 seconds annually.

In the linear function \( R(t) = 63.2 - 0.4t \), the number \(-0.4\) signifies this rate of change.
  • This negative rate means the world record time is getting shorter, which reflects improvement.
  • If the rate was different, say -0.3, it would mean the record improves by 0.3 seconds each year instead.
The rate of change is crucial as it quantifies the improvement over time. It's consistent and linear, suggesting a steady and predictable decrease in record time.
Modeling Linear Decrease
Modeling a linear decrease involves representing a scenario where a quantity reduces at a constant rate over time. In our example, the world record time is expressed through a linear function, capturing its steady decrease annually.

A linear model, like \( R(t) = 63.2 - 0.4t \), showcases exactly such a scenario. Here's how this model breaks down:
  • The initial value, 63.2 seconds, is the starting point, representing the record in 1950.
  • The constant term \(-0.4 \times t\) influences the decrease, indicating how much the record shortens annually.
However, while the model efficiently projects the records in the near future, it might not be valid indefinitely. Over time, physical limits will set a threshold, preventing indefinite improvements. This facet of linear models underscores their applicability over a realistic timeframe, rather than eternity.

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Most popular questions from this chapter

A roast is taken from the refrigerator (where it had been for several days) and placed immediately in a preheated oven to cook. The temperature \(R=R(t)\) of the roast \(t\) minutes after being placed in the oven is given by \(R=325-280 e^{-0.005 t}\) degrees Fahrenheit a. What is the temperature of the refrigerator? b. Express the temperature of the roast 30 minutes after being put in the oven in functional notation, and then calculate its value. c. By how much did the temperature of the roast increase during the first 10 minutes of cooking? d. By how much did the temperature of the roast increase from the first hour to 10 minutes after the first hour of cooking?

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Home equity: When you purchase a home by securing a mortgage, the total paid toward the principal is your equity in the home. The accompanying table shows the equity \(E\), in dollars, accrued after \(t\) years of payments on a mortgage of \(\$ 170,000\) at an APR of \(6 \%\) and a term of 30 years. $$ \begin{array}{|c|c|} \hline \begin{array}{c} t=\text { Years } \\ \text { of payments } \end{array} & \begin{array}{c} E=\text { Equity } \\ \text { in dollars } \end{array} \\ \hline 0 & 0 \\ \hline 5 & 11,808 \\ \hline 10 & 27,734 \\ \hline 15 & 49,217 \\ \hline 20 & 78,194 \\ \hline 25 & 117,279 \\ \hline 30 & 170,000 \\ \hline \end{array} $$ a. Explain the meaning of \(E(10)\) and give its value. b. Make a new table showing the average yearly rate of change in equity over each 5 -year period. c. Judging on the basis of your answer to part \(b\), does your equity accrue more rapidly early or late in the life of a mortgage? d. Use the average rate of change to estimate the equity accrued after 17 years. e. Would it make sense to use the average rate of change to estimate any function values beyond the limits of this table?

A stock market investment: A stock market investment of \(\$ 10,000\) was made in 1970 . During the decade of the \(1970 \mathrm{~s}\), the stock lost half its value. Beginning in 1980, the value increased until it reached \(\$ 35,000\) in 1990 . After that its value has remained stable. Let \(v=v(d)\) denote the value of the stock, in dollars, as a function of the date \(d\). a. What are the values of \(v(1970), v(1980)\), \(v(1990)\), and \(v(2000)\) ? b. Make a graph of \(v\) against \(d\). Label the axes appropriately. c. Estimate the time when your graph indicates that the value of the stock was most rapidly increasing.

A cold front: At 4 P.M. on a winter day, an arctic air mass moved from Kansas into Oklahoma, causing temperatures to plummet. The temperature \(T=T(h)\) in degrees Fahrenheit \(h\) hours after 4 P.M. in Stillwater, Oklahoma, on that day is recorded in the following table. $$ \begin{array}{|c|c|} \hline \begin{array}{c} h=\text { Hours } \\ \text { since } 4 \text { pM } \end{array} & T=\text { Temperature } \\ \hline 0 & 62 \\ \hline 1 & 59 \\ \hline 2 & 38 \\ \hline 3 & 26 \\ \hline 4 & 22 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \begin{array}{c} h=\text { Hours } \\ \text { since } 4 \text { pM } \end{array} & T=\text { Temperature } \\ \hline 0 & 62 \\ \hline 1 & 59 \\ \hline 2 & 38 \\ \hline 3 & 26 \\ \hline 4 & 22 \\ \hline \end{array} $$ a. Use functional notation to express the temperature in Stillwater at 5:30 P.M., and then estimate its value. b. What was the average rate of change per minute in temperature between 5 P.M. and 6 P.M.? What was the average decrease per minute over that time interval? c. Estimate the temperature at 5:12 P.M. d. At about what time did the temperature reach the freezing point? Explain your reasoning.
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