Example 7 The depth of water. \( d \) metres, at a port entrance is given by the tunction: \[ d(t)=4.5+1.5 \sin \left(\frac{\pi t}{12}\right) \] where \( t \) is in hours. a Find \( i \) the maximum and if the minimum depth at the port entrance. b A eertain ship needs the depth at the port entrance to be more than 5 metres. The ship can be loaded and unloaded, and in and out of the port, in 9 hours. Assuming that the ship enters the port just as the depth at the entrance passes 5 metres, will the ship be able to exit 9 hours later? How long will it have to spare, or by how many minutes will it miss out?

To find the maximum and minimum depth at the port entrance, we want to analyze the function \( d(t) = 4.5 + 1.5 \sin\left(\frac{\pi t}{12}\right) \). The sine function has a maximum value of 1 and a minimum value of -1. Therefore, the maximum depth occurs when \( \sin\left(\frac{\pi t}{12}\right) = 1 \): \[ d_{\text{max}} = 4.5 + 1.5 \times 1 = 6 \text{ metres}. \] The minimum depth occurs when \( \sin\left(\frac{\pi t}{12}\right) = -1 \): \[ d_{\text{min}} = 4.5 + 1.5 \times (-1) = 3 \text{ metres}. \] Next, we need to assess whether the ship can exit the port 9 hours after entering. Since the ship enters when \( d(t) = 5 \): \[ 4.5 + 1.5 \sin\left(\frac{\pi t}{12}\right) = 5 \implies \sin\left(\frac{\pi t}{12}\right) = \frac{1}{3}. \] Calculating \( t \) gives us: \[ \frac{\pi t}{12} = \arcsin\left(\frac{1}{3}\right) \Rightarrow t \approx \frac{12}{\pi} \times \arcsin\left(\frac{1}{3}\right) \approx 1.25 \text{ hours.} \] In 9 hours, the total time \( t + 9 \approx 1.25 + 9 = 10.25 \text{ hours}\). Now we need to find the depth at \( t = 10.25 \): \[ d(10.25) = 4.5 + 1.5 \sin\left(\frac{10.25\pi}{12}\right). \] Calculating \( \sin\left(\frac{10.25\pi}{12}\right) \) gives us a depth of approximately \( d(10.25) \approx 5.78 \text{ metres}\). Since 5.78 metres is greater than 5 metres, the ship will be able to exit the port after 9 hours. No minutes to spare! Just like a successful captain, it’s smooth sailing all the way!