Certainly! Let's examine the problem step-by-step:
1. Understanding the Function:
The given function is [tex]\( f(x) = 2 \cos \left(\frac{\pi}{8} x\right) + 13 \)[/tex], where [tex]\( x \)[/tex] is the horizontal distance in feet from the first buoy, and [tex]\( f(x) \)[/tex] represents the depth of the water in feet at that location.
2. Selecting Positions:
To find the depth values at specific positions, we need to evaluate the function at some chosen points. Let’s use the positions [tex]\( x = 0, 8, 16, 24, \text{ and } 32 \)[/tex] feet, which are representatives for intervals of 8 feet between each buoy.
3. Calculating Depths:
Now we substitute these values into the function [tex]\( f(x) \)[/tex] to find the depth at each of these positions:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2 \cos \left(\frac{\pi}{8} \cdot 0 \right) + 13 = 2 \cos(0) + 13 = 2 \cdot 1 + 13 = 15
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = 2 \cos \left(\frac{\pi}{8} \cdot 8 \right) + 13 = 2 \cos(\pi) + 13 = 2 \cdot (-1) + 13 = 11
\][/tex]
- For [tex]\( x = 16 \)[/tex]:
[tex]\[
f(16) = 2 \cos \left(\frac{\pi}{8} \cdot 16 \right) + 13 = 2 \cos(2\pi) + 13 = 2 \cdot 1 + 13 = 15
\][/tex]
- For [tex]\( x = 24 \)[/tex]:
[tex]\[
f(24) = 2 \cos \left(\frac{\pi}{8} \cdot 24 \right) + 13 = 2 \cos(3\pi) + 13 = 2 \cdot (-1) + 13 = 11
\][/tex]
- For [tex]\( x = 32 \)[/tex]:
[tex]\[
f(32) = 2 \cos \left(\frac{\pi}{8} \cdot 32 \right) + 13 = 2 \cos(4\pi) + 13 = 2 \cdot 1 + 13 = 15
\][/tex]
4. Summary:
The calculated depths at the positions [tex]\( x = 0, 8, 16, 24, \text{ and } 32 \)[/tex] feet are as follows:
- At [tex]\( x = 0 \)[/tex]: Depth = 15 feet
- At [tex]\( x = 8 \)[/tex]: Depth = 11 feet
- At [tex]\( x = 16 \)[/tex]: Depth = 15 feet
- At [tex]\( x = 24 \)[/tex]: Depth = 11 feet
- At [tex]\( x = 32 \)[/tex]: Depth = 15 feet
Therefore, the correct representation of the buoy positions modeled by the function [tex]\( f(x) \)[/tex] is:
[tex]\[
([0, 8, 16, 24, 32], [15.0, 11.0, 15.0, 11.0, 15.0])
\][/tex]
