The total income from a popular, newly released movie is given by \( f(t)=105.2+69 \ln (t) \) where \( f(t) \) is the income in millions of dollars \( t \) days after the opening. a) Estimate the total income 113 days after the opening day. Round to the nearest million dollars.

To find the total income 113 days after the opening day, we'll substitute \( t = 113 \) into the function \( f(t) = 105.2 + 69 \ln(t) \). First, calculate \( \ln(113) \): \[ \ln(113) \approx 4.827 \] Now plug this into the function: \[ f(113) = 105.2 + 69 \times 4.827 \] Calculating \( 69 \times 4.827 \): \[ 69 \times 4.827 \approx 332.943 \] Now add this to 105.2: \[ f(113) \approx 105.2 + 332.943 \approx 438.143 \] Finally, rounding to the nearest million dollars gives: \[ \text{Total income} \approx 438 \text{ million dollars.} \]