To determine the height of the tide after four hours given the function [tex]\( f(t) = 5 \cos \left(\frac{\pi}{6} t\right) + 7 \)[/tex], let's follow these detailed steps:
1. Understand the components of the function:
- The amplitude of the tide is 5 feet (this is the coefficient of the cosine function).
- The function's period is calculated based on the inside of the cosine function, i.e., [tex]\( \frac{\pi}{6} t \)[/tex]. Here, the period is [tex]\( \frac{2\pi}{\frac{\pi}{6}} = 12 \)[/tex] hours.
- The vertical shift is 7 feet, which means the entire cosine function is moved 7 feet upwards.
2. Evaluate the function at [tex]\( t = 4 \)[/tex] hours:
- First, calculate the angle when [tex]\( t = 4 \)[/tex]:
[tex]\[
\text{Angle} = \left(\frac{\pi}{6}\right) \times 4 = \frac{4\pi}{6} = \frac{2\pi}{3} \text{ radians}
\][/tex]
- Next, find the cosine of [tex]\( \frac{2\pi}{3} \)[/tex]:
[tex]\[
\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}
\][/tex]
- Now, insert this into the function:
[tex]\[
f(4) = 5 \cos \left(\frac{\pi}{6} \times 4 \right) + 7 = 5 \left( -\frac{1}{2} \right) + 7 = -2.5 + 7 = 4.5 \text{ feet}
\][/tex]
Therefore, after four hours, the height of the tide is [tex]\( 4.5 \)[/tex] feet.
Among the given options:
- 11.3 feet
- 9.5 feet
- 4.5 feet
- 2.6 feet
The correct answer is [tex]\( 4.5 \)[/tex] feet.
