High School

The water depth in a harbor rises and falls over time. The function [tex]f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7[/tex] models the water depth, in feet, after [tex]t[/tex] hours.

During the first 24 hours, at what times does the water depth reach a maximum?

A. at 5 and 17 hours
B. at 11 and 23 hours
C. at [tex]2, 8, 14[/tex], and 20 hours
D. at [tex]5, 11, 17[/tex], and 23 hours

Answer :

To determine when the water depth in the harbor reaches its maximum during the first 24 hours, we can analyze the function:

[tex]\[ f(t) = 4.1 \sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 \][/tex]

This function models the water depth in feet after [tex]\( t \)[/tex] hours. The sine function, [tex]\(\sin(\theta)\)[/tex], reaches its maximum value of 1. So, the maximum value of [tex]\( f(t) \)[/tex] occurs when:

[tex]\[ \sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1 \][/tex]

To find the values of [tex]\( t \)[/tex] that satisfy this equation within the first 24 hours, follow these steps:

1. Find the angle for which the sine is 1:

The sine of an angle is 1 at [tex]\(\frac{\pi}{2}\)[/tex] plus any integer multiple of [tex]\(2\pi\)[/tex]:

[tex]\[ \frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2k\pi \][/tex]

where [tex]\( k \)[/tex] is any integer.

2. Solve for [tex]\( t \)[/tex]:

Start with the principal value when [tex]\( k = 0 \)[/tex]:

[tex]\[ \frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} \][/tex]

Solve for [tex]\( t \)[/tex]:

Add [tex]\(\frac{\pi}{3}\)[/tex] to both sides:

[tex]\[ \frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} \][/tex]

Convert the right side to a common denominator:

[tex]\[ \frac{\pi}{6} t = \frac{3\pi}{6} + \frac{2\pi}{6} = \frac{5\pi}{6} \][/tex]

Multiply both sides by 6:

[tex]\[ \pi t = 5\pi \][/tex]

Divide both sides by [tex]\(\pi\)[/tex]:

[tex]\[ t = 5 \][/tex]

3. Determine additional times within the first 24 hours:

Since the period of the sine function [tex]\(\sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right)\)[/tex] is [tex]\( \frac{2\pi}{\frac{\pi}{6}} = 12 \)[/tex] hours, the water level reaches its maximum every 12 hours. Therefore, by adding the period or half-periods, we find more times:

- First maximum: [tex]\( t_1 = 5 \)[/tex] hours
- Second maximum: [tex]\( t_1 + 6 = 11 \)[/tex] hours
- Third maximum: [tex]\( t_1 + 12 = 17 \)[/tex] hours
- Fourth maximum: [tex]\( t_2 + 12 = 23 \)[/tex] hours

Therefore, during the first 24 hours, the water depth reaches a maximum at 5, 11, 17, and 23 hours.