Problem 68
Question
The number of hours the average person spends listening to satellite radio in year \(n\) is approximated by the sequence \(\left\\{b_{n}\right\\},\) where \(n=1\) corresponds to 2001 and \(b_{n}=945.54+\) 73.57 ln \(n .^{*}\) Round your final answers (not your calculations) to the following questions to the nearest hour. (a) For how many hours did the average person listen to satellite radio in 2002 and \(2006 ?\) (b) How much total time did the average person spend listening to satellite radio from 2001 to 2005 (inclusive)?
Step-by-Step Solution
Verified Answer
Additionally, how many total hours did the average person spend listening to satellite radio from 2001 to 2005?
Answer: The average person listened to satellite radio for approximately 1019 hours in 2002 and 1137 hours in 2006. They spent a total of 5283 hours listening to satellite radio from 2001 to 2005.
1Step 1: Find the hours in 2002 and 2006
We need to find the number of hours for 2002 (\(n = 2\)) and 2006 (\(n = 6\)). Plug these values into the given formula:
For 2002: \(b_2 = 945.54 + 73.57 \cdot \ln{2}\)
For 2006: \(b_6 = 945.54 + 73.57 \cdot \ln{6}\)
Now calculate the hours and round to the nearest hour:
2Step 2: Calculate the hours in 2002 and 2006
For 2002: \(b_2 = 945.54 + 73.57 \cdot \ln{2} \approx 1019\) hours
For 2006: \(b_6 = 945.54 + 73.57 \cdot \ln{6} \approx 1137\) hours
(a) The average person listened to satellite radio for 1019 hours in 2002 and 1137 hours in 2006.
3Step 3: Calculate the total hours from 2001 to 2005
To find the total hours from 2001 to 2005 (inclusive), we need to calculate and add the hours for \(n=1,2,3,4,5\):
Total hours \( = b_1 + b_2 + b_3 + b_4 + b_5 = (945.54 + 73.57 \cdot \ln{1}) + (945.54 + 73.57 \cdot \ln{2}) + (945.54 + 73.57 \cdot \ln{3})+(945.54 + 73.57 \cdot \ln{4})+(945.54 + 73.57 \cdot \ln{5})\)
4Step 4: Calculate and round the total hours
Calculate the sum of the hours and round the answer to the nearest hour:
Total hours \( \approx 945 + 1019 + 1070 + 1109 + 1140 = 5283\)
(b) The average person spent a total of 5283 hours listening to satellite radio from 2001 to 2005 (inclusive).
Key Concepts
Logarithmic FunctionsSequences and SeriesReal-world Applications of Precalculus
Logarithmic Functions
Logarithmic functions are one of the key concepts in mathematics, forming the inverse of exponential functions. They're written as \( y = \log_b(x) \) where \(\b\) is the base of the logarithm, \(x\) is the argument, and \(y\) is the logarithm of \(x\) to the base \(b\). Simply put, if \(b^y = x\), then \(y = \log_b(x)\).
The logarithm tells us how many times we need to multiply the base to get the argument. In the context of your textbook exercise, the natural logarithm (denoted as \(\ln\)) has a base \(e\), which is approximately 2.718. You might have encountered \(\ln(n)\) in the sequence \(b_n\), which denotes the number of hours spent listening to satellite radio.
To improve clarity for students, it's beneficial to focus on explaining what the function means. For instance, \(\ln(n)\) grows slower than \(n\) itself, which is key in understanding how the sequence evolves over time. Also, remind students that since these functions are continuous and increasing, the sequence values derived from them will also increase but at a decreasing rate.
The logarithm tells us how many times we need to multiply the base to get the argument. In the context of your textbook exercise, the natural logarithm (denoted as \(\ln\)) has a base \(e\), which is approximately 2.718. You might have encountered \(\ln(n)\) in the sequence \(b_n\), which denotes the number of hours spent listening to satellite radio.
Understanding \(\ln(n)\)
When working with natural logarithms, remember that \(\ln(1)\) is always 0 because \(e^0 = 1\). This 'zeroing out' of the logarithm at \(n=1\) is what prevented you from having to add anything extra to the first term of your sequence when calculating the total hours of radio listening from 2001 to 2005.To improve clarity for students, it's beneficial to focus on explaining what the function means. For instance, \(\ln(n)\) grows slower than \(n\) itself, which is key in understanding how the sequence evolves over time. Also, remind students that since these functions are continuous and increasing, the sequence values derived from them will also increase but at a decreasing rate.
Sequences and Series
Sequences and series are fundamental concepts in precalculus and beyond. A sequence is an ordered list of numbers generated according to some rule, and when we talk about series, we're referring to the sum of the elements of a sequence.
In the example provided, the sequence \( \{b_n\} \) is used to represent the hours spent listening to satellite radio. Each term \(b_n\) is determined by a logarithmic function of \(n\), creating a pattern that changes as \(n\) increases.
An improvement for helping students grasp this topic is to encourage them to see the bigger picture of summation notation and principles. Emphasize how individual terms are added to find the total and remind them that this process is fundamental to understanding series as a whole. Moreover, it might be useful to show the connection between the terms of the sequence and the corresponding values of \(n\), especially in the context of real-life timelines as illustrated by the satellite radio listening hours.
In the example provided, the sequence \( \{b_n\} \) is used to represent the hours spent listening to satellite radio. Each term \(b_n\) is determined by a logarithmic function of \(n\), creating a pattern that changes as \(n\) increases.
Calculating Terms and Totals
To find individual terms like \(b_2\) or \(b_6\), you simply plug the value of \(n\) into the function. When the problem asks for a total over several years, as in part (b) of your exercise, you're calculating the sum of a finite sequence, a concept which is integral to understanding series.An improvement for helping students grasp this topic is to encourage them to see the bigger picture of summation notation and principles. Emphasize how individual terms are added to find the total and remind them that this process is fundamental to understanding series as a whole. Moreover, it might be useful to show the connection between the terms of the sequence and the corresponding values of \(n\), especially in the context of real-life timelines as illustrated by the satellite radio listening hours.
Real-world Applications of Precalculus
Precalculus concepts aren't just for solving textbook exercises; they have numerous real-world applications. From physics and engineering to economics and social sciences, the principles of sequences, series, and logarithmic functions play crucial roles.
In the radio listening exercise, logarithmic sequences helped model the behavior of listeners over time, highlighting how such abstract mathematical concepts have practical uses. For example, logarithmic functions are widely used in finance to model compound interest, in biology to measure rates of decay or growth, and even in acoustics to understand sound intensity levels.
Students may be encouraged by knowing how these mathematical tools can help solve complex problems in diverse fields. This makes the learning experience relevant and shows the value of what might otherwise seem like purely academic concepts.
In the radio listening exercise, logarithmic sequences helped model the behavior of listeners over time, highlighting how such abstract mathematical concepts have practical uses. For example, logarithmic functions are widely used in finance to model compound interest, in biology to measure rates of decay or growth, and even in acoustics to understand sound intensity levels.
Connecting Math to the Real World
When explaining the significance of logarithmic sequences in precalculus, discuss their role in growth and decay models, which helps students appreciate the application of mathematics in various sectors. This understanding not only enriches their learning experience but also provides context for abstract concepts.Students may be encouraged by knowing how these mathematical tools can help solve complex problems in diverse fields. This makes the learning experience relevant and shows the value of what might otherwise seem like purely academic concepts.
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Problem 67
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