We are given that the temperature of a swimming pool is modeled by a trigonometric function. The highest temperature is $82^\circ$F and the lowest temperature is $76^\circ$F. The time it takes for the temperature to change between its extremes is 12 hours. We need to find the equation that models the temperature of the pool as a function of time in hours, given that the pool begins at $82^\circ$F.

1. Problem Description

We are given that the temperature of a swimming pool is modeled by a trigonometric function. The highest temperature is $82^\circ$F and the lowest temperature is $76^\circ$F. The time it takes for the temperature to change between its extremes is 12 hours. We need to find the equation that models the temperature of the pool as a function of time in hours, given that the pool begins at $82^\circ$F.

2. Solution Steps

The general form of a cosine function is:

$y = A \cos(B(t - C)) + D$

where:

$A$ is the amplitude,

$B$ is related to the period,

$C$ is the horizontal shift,

$D$ is the vertical shift.

The amplitude $A$ is half the difference between the highest and lowest temperatures:

$A = \frac{82 - 76}{2} = \frac{6}{2} = 3$

The vertical shift $D$ is the average of the highest and lowest temperatures:

$D = \frac{82 + 76}{2} = \frac{158}{2} = 79$

The period is the time it takes for the temperature to complete one full cycle (from high to low and back to high). Since it takes 12 hours to change between extremes (high to low), the full cycle (high to high) will take $2 \times 12 = 24$ hours. Thus, the period $P = 24$.

The relationship between $B$ and the period $P$ is:

$B = \frac{2\pi}{P} = \frac{2\pi}{24} = \frac{\pi}{12}$

Since the pool begins at its highest temperature ($82^\circ$F), we can use a cosine function without a horizontal shift (C = 0). Therefore, the equation is:

$y = 3 \cos\left(\frac{\pi}{12}t\right) + 79$

or

$y = 3 \cos\left(\frac{2\pi}{24}t\right) + 79$

3. Final Answer

$y = 3 \cos(\frac{2\pi}{24}t) + 79$