the water level at a pier is modeled by the function ( y = 2.5cosleft(\frac{2pi}{12.5}x\right)+12 ), where (…

the water level at a pier is modeled by the function ( y = 2.5cosleft(\frac{2pi}{12.5}x\right)+12 ), where ( y ) represents the water level measured in meters, and ( x ) represents the number of hours since the last high tide. after how many hours is the water first expected to reach a depth of 12 meters? round to the nearest tenth of an hour.\n1.6 hours\n3.1 hours\n14.4 hours\n19.6 hours

the water level at a pier is modeled by the function ( y = 2.5cosleft(\frac{2pi}{12.5}x\right)+12 ), where ( y ) represents the water level measured in meters, and ( x ) represents the number of hours since the last high tide. after how many hours is the water first expected to reach a depth of 12 meters? round to the nearest tenth of an hour.\n1.6 hours\n3.1 hours\n14.4 hours\n19.6 hours

Answer

Explanation:

Step1: Substitute (y = 12) into the equation

Substitute (y = 12) into (y=2.5\cos(\frac{2\pi}{12.5}x)+12). We get (12 = 2.5\cos(\frac{2\pi}{12.5}x)+12). Subtract 12 from both sides: (0 = 2.5\cos(\frac{2\pi}{12.5}x)). Then (\cos(\frac{2\pi}{12.5}x)=0).

Step2: Solve for (x)

We know that (\cos\theta = 0) when (\theta=\frac{\pi}{2}+ 2k\pi) or (\theta=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}). For the first - non - negative solution, we take (\theta=\frac{\pi}{2}) (since we want the first time (x>0)). Set (\frac{2\pi}{12.5}x=\frac{\pi}{2}). Multiply both sides by (\frac{12.5}{2\pi}): (x=\frac{12.5}{4}). Calculate (x = 3.125\approx3.1).

Answer:

3.1 hours