Questions: The equation y=2(3^t) shows the number of infected people from an outbreak of the Norovirus. 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 1,458? a.) 5 weeks b.) 10 weeks c.) 6 weeks d.) 12 weeks

The equation y=2(3^t) shows the number of infected people from an outbreak of the Norovirus. 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 1,458? a.) 5 weeks b.) 10 weeks c.) 6 weeks d.) 12 weeks
Transcript text: The equation $y=2\left(3^{t}\right)$ shows the number of infected people from an outbreak of the Norovirus. 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 $1,458 ?$ a.) 5 weeks b.) 10 weeks c.) 6 weeks d.) 12 weeks
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Solution

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Solution Steps

Step 1: Start with the general equation

Given the exponential growth model, the equation is \(y = a \cdot b^t\), where:

  • \(y\) is the quantity of interest (number of infected people),
  • \(a\) is the initial quantity at \(t = 0\),
  • \(b\) is the growth factor per time unit,
  • \(t\) is the time in appropriate units (weeks).
Step 2: Solve for \(t\) when \(y\) is given

Rearrange the equation to solve for \(t\): \(t = \frac{\log(y/a)}{\log(b)}\).

Step 3: Substitute the given values into the equation

Substitute \(a = 2\), \(b = 3\), and \(y = 1458\) into the equation to get \(t\).

Step 4: Calculate \(t\)

Using the values, \(t = \frac{\log(1458/2)}{\log(3)} = 6\).

Final Answer:

The time \(t\) when the quantity of interest \(y\) is reached is approximately 6 weeks.

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