Questions: The equation y=20 * 3^? 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.) 3.56 weeks b.) 2.45 weeks c.) 2.88 weeks d.) 3.24 weeks

Solution Steps

To solve the exponential equation \( y = 20 \cdot 3^t \) for \( t \) when \( y = 1000 \), we need to isolate \( t \). This can be done by taking the logarithm of both sides of the equation. By applying the properties of logarithms, we can solve for \( t \) as follows:

1. Set the equation to \( 1000 = 20 \cdot 3^t \).
2. Divide both sides by 20 to isolate the exponential term: \( 50 = 3^t \).
3. Take the logarithm of both sides: \( \log(50) = \log(3^t) \).
4. Use the logarithmic identity \(\log(a^b) = b \cdot \log(a)\) to simplify: \( \log(50) = t \cdot \log(3) \).
5. Solve for \( t \) by dividing both sides by \(\log(3)\).

We start with the equation representing the number of infected people from an outbreak of whooping cough: \[ y = 20 \cdot 3^t \] We need to find \( t \) when \( y = 1000 \).

Substituting \( y = 1000 \) into the equation gives: \[ 1000 = 20 \cdot 3^t \] Dividing both sides by 20, we have: \[ 50 = 3^t \]

Taking the logarithm of both sides, we get: \[ \log(50) = \log(3^t) \] Using the property of logarithms, this simplifies to: \[ \log(50) = t \cdot \log(3) \]

Rearranging the equation to solve for \( t \) gives: \[ t = \frac{\log(50)}{\log(3)} \] Calculating this yields: \[ t \approx 3.5609 \]

Final Answer