To solve this problem, let’s consider the function [tex]\( y = -\cos(x) + 7 \)[/tex] and its properties.
### Step 1: Understanding the Cosine Function
The cosine function, [tex]\( \cos(x) \)[/tex]:
- is periodic with a period of [tex]\( 2\pi \)[/tex]. This means it repeats every [tex]\( 2\pi \)[/tex] units along the [tex]\( x \)[/tex]-axis.
- oscillates between -1 and 1.
### Step 2: Transforming the Cosine Function
The given function is:
[tex]\[ y = -\cos(x) + 7 \][/tex]
By noting the transformation:
- [tex]\(-\cos(x)\)[/tex] reflects [tex]\( \cos(x) \)[/tex] vertically over the x-axis.
- Adding 7 shifts the graph upward by 7 units.
This means the new function oscillates between 6 and 8.
### Step 3: Analyzing the Period of the Function
The period of the cosine function itself is [tex]\( 2\pi \)[/tex]. This period remains unchanged because there are no factors multiplying [tex]\( x \)[/tex] that would stretch or compress the graph horizontally.
### Step 4: Determining the Number of Cycles in 60 Feet
To find out how many cycles of the function fit into a 60 feet distance, we need to figure out how many periods of [tex]\( 2\pi \)[/tex] fit into 60 feet:
[tex]\[ \text{number of cycles} = \frac{60}{2\pi} \][/tex]
Given the numerical result:
[tex]\[ \text{number of cycles} = 9.54929658551372 \][/tex]
This means the function completes approximately 9.55 cycles in a 60 feet distance.
### Step 5: Finding the Number of Returns to the Wall
In each full cycle of the cosine function (one period), the graph passes through its starting point twice — once going up and once going down. This also applies to our transformed function [tex]\( y = -\cos(x) + 7 \)[/tex].
Therefore, within each cycle, the gum will return to the wall twice. To find the total number of returns to the wall, we double the number of cycles:
[tex]\[ \text{returns to wall} = 2 \times \text{number of cycles} \][/tex]
Given the numerical results:
[tex]\[ \text{returns to wall} = 2 \times 9.54929658551372 = 19.0985931710274 \][/tex]
### Final Result
To sum up, as the gum travels a distance of 60 feet:
- It completes approximately 9.55 cycles.
- It returns to the wall approximately 19.10 times.
