The beam from a lighthouse completes one rotation every 2 min. At time \( t \), the distance \( d \) shown in the following figure is \[ d(t)=3 \text { tan }(\pi t) \] where \( t \) is measured in minutes and \( d \) in miles. \( \begin{array}{r}\text { (a) } \begin{array}{r}\text { Find } d(0.1), ~ \\ \text { (a) }\end{array} \\ d(0.25) \text {, and } d(0.35) \text { (in miles). (Round your answers to two decimal places.) } \\ d(0.25) \\ d(0.35)\end{array} \)

To find the values of \( d(0.1) \), \( d(0.25) \), and \( d(0.35) \), we can substitute these values of \( t \) into the formula \( d(t) = 3 \tan(\pi t) \). 1. For \( t = 0.1 \): \[ d(0.1) = 3 \tan(\pi \times 0.1) = 3 \tan(0.31416) \approx 3 \times 0.32175 \approx 0.96524 \text{ miles} \approx 0.97 \text{ miles (rounded to two decimal places)} \] 2. For \( t = 0.25 \): \[ d(0.25) = 3 \tan(\pi \times 0.25) = 3 \tan(0.78540) \approx 3 \times 1 \approx 3.00 \text{ miles (exact)} \] 3. For \( t = 0.35 \): \[ d(0.35) = 3 \tan(\pi \times 0.35) = 3 \tan(1.09956) \approx 3 \times 1.42815 \approx 4.28445 \text{ miles} \approx 4.28 \text{ miles (rounded to two decimal places)} \] So the results are: - \( d(0.1) \approx 0.97 \) miles - \( d(0.25) = 3.00 \) miles - \( d(0.35) \approx 4.28 \) miles