Question
The equation \( y=20 \cdot 3^{t} \) shows the number of infected people from an outbreak of whooping cough. The variable \( y \) represents the number of infected people, and \( t \) represents time in weeks. In how many weeks will the number of infected people reach 1,000 ? a.) 2.45 weeks b.) 2.88 weeks c.) 3.56 weeks d.) 3.24 weeks
Ask by Mejia Hodgson. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The number of infected people will reach 1,000 in approximately 3.56 weeks.
Solution
To solve for t when y = 1,000, start with the equation:
20 · 3^t = 1,000
First, isolate the exponential term:
3^t = 1,000 / 20 = 50
Next, take the natural logarithm of both sides:
ln(3^t) = ln(50)
Use the logarithm power rule:
t · ln(3) = ln(50)
Now, solve for t:
t = ln(50) / ln(3)
Calculating the values:
ln(50) ≈ 3.912
ln(3) ≈ 1.099
Thus:
t ≈ 3.912 / 1.099 ≈ 3.56
The number of weeks required for the number of infected people to reach 1,000 is approximately 3.56 weeks, which corresponds to option c.
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Simplify this solution Extra Insights
To find out when the number of infected people will reach 1,000, we will set up the equation: \[ 1000 = 20 \cdot 3^t \] First, divide both sides by 20: \[ 50 = 3^t \] Now, we can solve for \( t \) by taking the logarithm of both sides: \[ t = \log_3(50) \] Using the change of base formula, this can be calculated as: \[ t = \frac{\log_{10}(50)}{\log_{10}(3)} \] Calculating the logarithms: - \( \log_{10}(50) \approx 1.6990 \) - \( \log_{10}(3) \approx 0.4771 \) Now, divide those values: \[ t \approx \frac{1.6990}{0.4771} \approx 3.56 \] Thus, \( t \) is approximately 3.56 weeks. So, the answer is c.) 3.56 weeks.
