Mustafa is staring at a pizza spinning on a wheel in a glass case.

The pizza is cut into four even slices. The distance \[P(t)\] (in \[\text{cm}\]) between the center of the tastiest looking slice and the door of the glass case as a function of time \[t\] (in seconds) can be modeled by a sinusoidal function of the form \[a\cdot\cos(b\cdot t)+d\].

At \[t=0\], the center of the tastiest looking slice is farthest from the door, at a distance of \[30\text{ cm}\] away. After \[2\pi\] seconds, it is closest to the door, at a distance of \[10\text{ cm}\].

**Find \[P(t)\].**
*\[\text{\it{t}}\] should be in radians.*

\[P(t) = \] 

We are asked to find the function \( P(t) \) that models the distance between the center of the tastiest slice of pizza and the door of the glass case as a function of time \( t \). The function is of the form:

\[
P(t) = a \cdot \cos(b \cdot t) + d
\]

### Step 1: Identify \(d\), the vertical shift.
The problem states that at \( t = 0 \), the distance is at its maximum, \( 30 \text{ cm} \), and after \( 2\pi \) seconds, the distance is at its minimum, \( 10 \text{ cm} \). The average of the maximum and minimum values gives the vertical shift \( d \):

\[
d = \frac{30 \text{ cm} + 10 \text{ cm}}{2} = \frac{40 \text{ cm}}{2} = 20 \text{ cm}
\]

### Step 2: Determine the amplitude \( a \).
The amplitude \( a \) is the difference between the maximum value and the average value:

\[
a = 30 \text{ cm} - 20 \text{ cm} = 10 \text{ cm}
\]

### Step 3: Determine the period and frequency \( b \).
The problem states that after \( 2\pi \) seconds, the slice is at its minimum distance, meaning one full cycle has occurred. Since the period \( T \) is \( 2\pi \) seconds, the angular frequency \( b \) is:

\[
b = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1
\]

### Step 4: Determine the phase shift.
At \( t = 0 \), \( P(t) \) is at its maximum, which corresponds to a cosine function without any phase shift, meaning the phase shift is \( 0 \).

### Step 5: Write the equation.
Combining all the above information, we get:

\[
P(t) = 10 \cdot \cos(1 \cdot t) + 20
\]

### Final Answer:
\[
P(t) = 10 \cdot \cos(t) + 20
\]

Would you like more details on any part of the solution, or do you have any other questions?

Here are some related questions to explore further:
1. What would the function look like if the pizza was spinning faster?
2. How would the equation change if the minimum distance was \(15\) cm instead of \(10\) cm?
3. Can you find the time when the slice is exactly halfway between its maximum and minimum distances?
4. What would the function be if the distance was measured from the closest point instead of the farthest?
5. How does the period of the cosine function affect the speed of the pizza's rotation?

**Tip:** When working with sinusoidal functions, always verify the amplitude, period, and vertical shift, as they are critical in defining the behavior of the function.

Learn how to model the distance of the tastiest slice of pizza from the glass case door using a sinusoidal function. Understand the amplitude, vertical shift, period, and phase shift in the context of trigonometry and sinusoidal functions. Suitable for high school students and those interested in applied mathematics.

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