The factory sales of cell phones from 2000 to 2005 can be modeled by the function 116(2) 0.018t where t=0 in 2000 and S represents the sales in millions of S dollars. = A) Assuming that the sales of cell phones continued to increase at the same rate, find the factory sales of cell phones in 2021 to the nearest million. Explain steps and your reasoning. What does your answer mean in terms of the problem? B) According to the function, when would the sales reach 200 million? Explain your steps and reasoning. What does your answer mean in terms of the problem?
The factory sales of cell phones from 2000 to 2005 can be modeled by the function 116(2) 0.018t where t=0 in 2000 and S represents the sales in millions of S dollars. = A) Assuming that the sales of cell phones continued to increase at the same rate, find the factory sales of cell phones in 2021 to the nearest million. Explain steps and your reasoning. What does your answer mean in terms of the problem? B) According to the function, when would the sales reach 200 million? Explain your steps and reasoning. What does your answer mean in terms of the problem?
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
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![The factory sales of cell phones from 2000 to 2005 can be modeled by the function
\[ S = 116 (2)^{0.018t} \]
where \( t = 0 \) in 2000 and \( S \) represents the sales in millions of dollars.
**A) Assuming that the sales of cell phones continued to increase at the same rate, find the factory sales of cell phones in 2021 to the nearest million. Explain steps and your reasoning. What does your answer mean in terms of the problem?**
To find the factory sales in 2021:
1. Calculate the number of years from 2000 to 2021: \( t = 2021 - 2000 = 21 \).
2. Substitute \( t = 21 \) into the function: \( S = 116 (2)^{0.018 \times 21} \).
3. Calculate the exponent: \( 0.018 \times 21 = 0.378 \).
4. Compute the value of \( 2^{0.378} \).
5. Multiply the result by 116 to get the sales in millions of dollars.
Thus, the factory sales in 2021 can be calculated. This calculation will provide the projected sales for the year 2021 if the growth rate remains consistent.
**B) According to the function, when would the sales reach 200 million? Explain your steps and reasoning. What does your answer mean in terms of the problem?**
To find when the sales would reach 200 million:
1. Set \( S = 200 \) in the function: \( 200 = 116 (2)^{0.018t} \).
2. Solve for \( t \):
\[ \frac{200}{116} = (2)^{0.018t} \]
\[ 1.724 = (2)^{0.018t} \]
3. Take the natural logarithm of both sides to solve for the exponent:
\[ \ln(1.724) = 0.018t \cdot \ln(2) \]
4. Divide both sides by \( \ln(2) \):
\[ t = \frac{\ln(1.724)}{0.018 \cdot \ln(2)} \]
5. Calculate the value of \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9d14dc9-6899-40d3-8d6b-066b6648b231%2Fe3eaeee9-4c43-43fe-9d2c-49dc9b02ef48%2Fi08y3e_processed.png&w=3840&q=75)
Transcribed Image Text:The factory sales of cell phones from 2000 to 2005 can be modeled by the function
\[ S = 116 (2)^{0.018t} \]
where \( t = 0 \) in 2000 and \( S \) represents the sales in millions of dollars.
**A) Assuming that the sales of cell phones continued to increase at the same rate, find the factory sales of cell phones in 2021 to the nearest million. Explain steps and your reasoning. What does your answer mean in terms of the problem?**
To find the factory sales in 2021:
1. Calculate the number of years from 2000 to 2021: \( t = 2021 - 2000 = 21 \).
2. Substitute \( t = 21 \) into the function: \( S = 116 (2)^{0.018 \times 21} \).
3. Calculate the exponent: \( 0.018 \times 21 = 0.378 \).
4. Compute the value of \( 2^{0.378} \).
5. Multiply the result by 116 to get the sales in millions of dollars.
Thus, the factory sales in 2021 can be calculated. This calculation will provide the projected sales for the year 2021 if the growth rate remains consistent.
**B) According to the function, when would the sales reach 200 million? Explain your steps and reasoning. What does your answer mean in terms of the problem?**
To find when the sales would reach 200 million:
1. Set \( S = 200 \) in the function: \( 200 = 116 (2)^{0.018t} \).
2. Solve for \( t \):
\[ \frac{200}{116} = (2)^{0.018t} \]
\[ 1.724 = (2)^{0.018t} \]
3. Take the natural logarithm of both sides to solve for the exponent:
\[ \ln(1.724) = 0.018t \cdot \ln(2) \]
4. Divide both sides by \( \ln(2) \):
\[ t = \frac{\ln(1.724)}{0.018 \cdot \ln(2)} \]
5. Calculate the value of \( t \).
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