Questions: The equation (y=4 cdot 3^t) shows the number of infected people from an outbreak of the measles. 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 600?

Solution Steps

Given the equation \( y = 4 \cdot 3^{t} \), we need to find the value of \( t \) when \( y = 600 \). Substitute \( y = 600 \) into the equation: \[ 600 = 4 \cdot 3^{t} \]

Divide both sides of the equation by 4 to isolate the exponential term: \[ \frac{600}{4} = 3^{t} \] \[ 150 = 3^{t} \]

Take the natural logarithm (ln) of both sides to solve for \( t \): \[ \ln(150) = \ln(3^{t}) \] Using the logarithm power rule \( \ln(a^{b}) = b \cdot \ln(a) \), the equation becomes: \[ \ln(150) = t \cdot \ln(3) \] Now, solve for \( t \): \[ t = \frac{\ln(150)}{\ln(3)} \]

Compute the value of \( t \) using a calculator: \[ t \approx \frac{5.0106}{1.0986} \approx 4.56 \] Thus, the number of weeks required for the number of infected people to reach 600 is approximately 4.56 weeks.

Final Answer