Problem 17
Question
Chicago's average monthly rainfall, \(R=f(t)\) inches, is given as a function of month, \(t,\) in Table 4.7 . (January is \(t=1 .\) ) Solve and interpret: (a) \(\quad f(t)=3.7\) (b) \(\quad f(t)=f(2)\) $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline t & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline R & 1.8 & 1.8 & 2.7 & 3.1 & 3.5 & 3.7 & 3.5 & 3.4 \\ \hline \end{array} $$
Step-by-Step Solution
Verified Answer
Answer: January and February.
1Step 1: Interpret the function
The function \(R = f(t)\) represents the average monthly rainfall in Chicago with respect to the month \(t\). In our problem, we have to find the month that corresponds to a given amount of rainfall based on the data provided in the table.
### Step 2: Solve 'f(t) = 3.7' ###
2Step 2: Solve for f(t) = 3.7
To find the month for which the average rainfall is 3.7 inches, we look at the table and search for a month where \(R = 3.7\). We see that in June (\(t = 6\)), the average rainfall is 3.7 inches.
### Step 3: Interpret the result of 'f(t) = 3.7' ###
3Step 3: Interpret the result of f(t) = 3.7
Since we found that the average rainfall is 3.7 inches in June, we can say that the month when the average rainfall is 3.7 inches is June.
### Step 4: Solve 'f(t) = f(2)' ###
4Step 4: Solve for f(t) = f(2)
To find the months where the average rainfall is equal to the average rainfall in February, we look for months where \(f(t) = f(2)\). According to the table, in February (\(t = 2\)), the average rainfall is 1.8 inches. Therefore, we are looking for months where \(f(t) = 1.8\).
5Step 5: Find months with R = 1.8
Looking at the table, we see that the average rainfall is 1.8 inches in January and February, which correspond to \(t = 1\) and \(t = 2\).
### Step 5: Interpret the result of 'f(t) = f(2)' ###
6Step 6: Interpret the result of f(t) = f(2)
Since we found that the average rainfall is 1.8 inches in both January and February, the months with average rainfall equal to that of February are January and February.
Key Concepts
Function InterpretationTable AnalysisEquivalent Rainfall Values
Function Interpretation
Functions are a way to describe how one quantity depends on another. In our exercise, the function \(R = f(t)\) provides insight into Chicago's average monthly rainfall. Here, \(R\) stands for the rainfall in inches, and \(t\) represents the month (with January as \(t=1\), February as \(t=2\), and so on). This function helps us understand the relationship between the month and the average rainfall.
To use this function, we interpret \(f(t)\) as the rainfall value for a specific month. By analyzing the given table, we learn how to read these values, such as seeing which month corresponds to a specific rainfall amount. This interpretation helps us solve equations like \(f(t) = 3.7\), where we need to identify the month for this given rainfall amount.
To use this function, we interpret \(f(t)\) as the rainfall value for a specific month. By analyzing the given table, we learn how to read these values, such as seeing which month corresponds to a specific rainfall amount. This interpretation helps us solve equations like \(f(t) = 3.7\), where we need to identify the month for this given rainfall amount.
- Function \(R=f(t)\) is unique to Chicago and its rainfall pattern.
- It offers a way to forecast or understand what to expect during different times of the year.
Table Analysis
Understanding data in a table format is crucial to solving problems related to functions like \(R=f(t)\). The table in this exercise lists months and their corresponding average rainfall values, allowing us to analyze rainfall patterns across different months.
Each entry in the table has a month, labeled \(t\), and its associated rainfall value, \(R\). By focusing on different values of \(R\), we can determine the relationship each has with the respective month. The problem then becomes a matter of filtering this data for specific conditions, like when \(R=3.7\).
Each entry in the table has a month, labeled \(t\), and its associated rainfall value, \(R\). By focusing on different values of \(R\), we can determine the relationship each has with the respective month. The problem then becomes a matter of filtering this data for specific conditions, like when \(R=3.7\).
- Switching between month numbers and rain amounts is a key part of finding solutions.
- Scanning the table for specific values helps identify trends, such as common rainfall values.
Equivalent Rainfall Values
Working with equivalent rainfall values means comparing different months that share the same rainfall measurements. In this exercise, we were asked to find months where the rainfall was equal to specific values.
For example, by consulting the table, we discovered that \(f(t) = 3.7\) led us to June. We then determined months where rainfall equaled February's, which was 1.8 inches, finding both January and February matched this.
For example, by consulting the table, we discovered that \(f(t) = 3.7\) led us to June. We then determined months where rainfall equaled February's, which was 1.8 inches, finding both January and February matched this.
- This process illustrates how some months might share the same patterns or outputs.
- Equivalent values highlight similarities in data, enabling factual comparisons.
- Such comparisons are beneficial for recognizing repetitive weather patterns.
Other exercises in this chapter
Problem 17
In Exercises 14-17, assume the two quantities are directly proportional to each other. If \(s=35\) when \(t=25,\) find \(t\) when \(s\) is 14 .
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In Exercises \(17-20,\) which letters stand for variables and which for constants? $$ V(r)=(4 / 3) \pi r^{2} $$
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Evaluate the expressions given that $$ h(t)=10-3 t $$ $$ h(k-3) $$
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