Graph of \( f \) At a coastal city, the height of the tide, in feet (ft), is modeled by the function \( h \), defined by \( h(t)=6.3 \cos \left(\frac{\pi}{6} t\right)+7.5 \) for \( 0 \leq t \leq 12 \) hours. Based on the model, which of the following is true? (A) The maximum height of the tide is 13.8 ft . (B) The maximum height of the tide occurs at \( t=6 \) hours. (C) The minimum height of the tide is 1 ft . (D) The minimum height of the tide occurs at \( t=12 \) hours. (D)

To analyze the function \( h(t) = 6.3 \cos \left(\frac{\pi}{6} t\right) + 7.5 \), we can determine the maximum and minimum heights of the tide, as well as when they occur. 1. **Maximum Height**: The cosine function \( \cos(x) \) has a maximum value of 1. Therefore, the maximum height of the tide occurs when \( \cos \left(\frac{\pi}{6} t\right) = 1 \): \[ h_{\text{max}} = 6.3 \cdot 1 + 7.5 = 6.3 + 7.5 = 13.8 \text{ ft} \] So, statement (A) is true. 2. **Minimum Height**: The minimum value of the cosine function is -1. Therefore, the minimum height of the tide occurs when \( \cos \left(\frac{\pi}{6} t\right) = -1 \): \[ h_{\text{min}} = 6.3 \cdot (-1) + 7.5 = -6.3 + 7.5 = 1.2 \text{ ft} \] So, statement (C) is false because the minimum height is 1.2 ft, not 1 ft. 3. **When Maximum Height Occurs**: The maximum height occurs when \( \cos \left(\frac{\pi}{6} t\right) = 1 \). This happens when: \[ \frac{\pi}{6} t = 2k\pi \quad \text{for integers } k \] Solving for \( t \): \[ t = 12k \] For \( k = 0 \), \( t = 0 \) hours. For \( k = 1 \), \( t = 12 \) hours. Thus, the maximum height occurs at \( t = 0 \) hours and \( t = 12 \) hours, not at \( t = 6 \) hours. Therefore, statement (B) is false. 4. **When Minimum Height Occurs**: The minimum height occurs when \( \cos \left(\frac{\pi}{6} t\right) = -1 \). This happens when: \[ \frac{\pi}{6} t = (2k + 1)\pi \quad \text{for integers } k \] Solving for \( t \): \[ t = 6(2k + 1) \] For \( k = 0 \), \( t = 6 \) hours. For \( k = 1 \), \( t = 18 \) hours (which is outside the given range). Thus, the minimum height occurs at \( t = 6 \) hours, not at \( t = 12 \) hours. Therefore, statement (D) is false. In summary: - (A) True: The maximum height is 13.8 ft. - (B) False: The maximum height does not occur at \( t = 6 \) hours. - (C) False: The minimum height is 1.2 ft, not 1 ft. - (D) False: The minimum height does not occur at \( t = 12 \) hours. Thus, the only true statement is (A).