To determine the depth of water at high tide modeled by the function [tex]\( y = 3 \sin \left(\frac{\pi}{6} x\right) + 7 \)[/tex], we need to find the maximum value of this function.
1. Understanding the Function:
- The function given is [tex]\( y = 3 \sin \left(\frac{\pi}{6} x\right) + 7 \)[/tex].
- This is a sinusoidal function where [tex]\( 3 \)[/tex] is the amplitude, meaning the maximum deviation from the central value (in this case 7) due to the sine function.
2. Amplitude and Midline:
- The term [tex]\( 3 \sin \left(\frac{\pi}{6} x\right) \)[/tex] determines how much the sine wave oscillates.
- The [tex]\( +7 \)[/tex] shifts the entire sine wave up by 7 units.
3. Maximum Value of Sine:
- The sine function [tex]\( \sin \left( \cdot \right) \)[/tex] oscillates between -1 and 1.
- Therefore, the maximum value of [tex]\( \sin \left( \frac{\pi}{6} x \right) \)[/tex] is 1.
4. Calculating the Maximum Depth:
- Substitute the maximum value of [tex]\( \sin \left( \frac{\pi}{6} x \right) = 1 \)[/tex] back into the equation:
[tex]\[
y = 3 \cdot 1 + 7
\][/tex]
[tex]\[
y = 3 + 7
\][/tex]
[tex]\[
y = 10
\][/tex]
The maximum possible depth of the water at high tide is [tex]\( 10 \)[/tex] feet.
