Answer :
To tackle this problem, let's proceed step-by-step.
1. Understanding the Function [tex]\( y = -\cos(x) + 7 \)[/tex]:
- This function describes a cosine wave that has been vertically shifted up by 7 units and vertically flipped. The usual cosine function, [tex]\( \cos(x) \)[/tex], varies between -1 and 1. When negated, [tex]\( -\cos(x) \)[/tex] varies between -1 and 1 as well but with a switch in sign; it will go from 1 to -1.
- Adding 7 to [tex]\( -\cos(x) \)[/tex] moves the entire range up by 7 units. Thus, [tex]\( -\cos(x) + 7 \)[/tex] will vary between 6 and 8.
2. Basic Characteristics:
- The period of [tex]\( \cos(x) \)[/tex] is [tex]\( 2\pi \)[/tex]. Given the transformation of flipping and shifting, the period remains [tex]\( 2\pi \)[/tex]. This means that the pattern of the wave repeats every [tex]\( 2\pi \)[/tex] units along the x-axis.
3. Finding when the Gum Returns to the Wall:
- The "wall" in this case corresponds to [tex]\( y = 7 \)[/tex].
- We need to find where the graph of [tex]\( y = -\cos(x) + 7 \)[/tex] intersects the line [tex]\( y = 7 \)[/tex].
4. Solving [tex]\( -\cos(x) + 7 = 7 \)[/tex]:
- Set [tex]\( y = -\cos(x) + 7 \)[/tex] equal to 7:
[tex]\[ -\cos(x) + 7 = 7 \][/tex]
Subtract 7 from both sides:
[tex]\[ -\cos(x) = 0 \][/tex]
Multiply both sides by -1:
[tex]\[ \cos(x) = 0 \][/tex]
- The cosine of [tex]\( x \)[/tex] equals 0 at:
[tex]\[ x = \frac{\pi}{2} + k\pi \quad \text{for integer } k \][/tex]
This is because [tex]\( \cos\left(\frac{\pi}{2}\right) = 0 \)[/tex], and cosine has a period of [tex]\( \pi \)[/tex] for subsequent zeros.
5. Identifying All Intersections within 60 Feet:
- We need to find all [tex]\( x \)[/tex] values where [tex]\( \cos(x) = 0 \)[/tex] and [tex]\( 0 \leq x \leq 60 \)[/tex]:
[tex]\[ x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots, \frac{\pi}{2} + n\pi \][/tex]
- The maximum value of [tex]\( n \)[/tex] where [tex]\( x \leq 60 \)[/tex]:
[tex]\[ \frac{\pi}{2} + n\pi \leq 60 \][/tex]
Substitute the [tex]\( \frac{\pi}{2} \)[/tex] to a common term:
[tex]\[ \pi(n + \frac{1}{2}) \leq 60 \][/tex]
Divide both sides by [tex]\( \pi \)[/tex]:
[tex]\[ n + \frac{1}{2} \leq \frac{60}{\pi} \][/tex]
Approximately (since [tex]\( \pi \approx 3.14159265359 \)[/tex]):
[tex]\[ n + \frac{1}{2} \leq 19.0986 \][/tex]
Subtract [tex]\( \frac{1}{2} \)[/tex] from both sides:
[tex]\[ n \leq 18.5986 \][/tex]
The largest integer [tex]\( n \leq 18.5986 \)[/tex] is 18.
Thus, [tex]\( n \)[/tex] can take on the values from 0 to 18 inclusive, giving us 19 points since it includes both 0 and 18.
So, the gum returns to the wall 19 times as it travels a distance of 60 feet.
1. Understanding the Function [tex]\( y = -\cos(x) + 7 \)[/tex]:
- This function describes a cosine wave that has been vertically shifted up by 7 units and vertically flipped. The usual cosine function, [tex]\( \cos(x) \)[/tex], varies between -1 and 1. When negated, [tex]\( -\cos(x) \)[/tex] varies between -1 and 1 as well but with a switch in sign; it will go from 1 to -1.
- Adding 7 to [tex]\( -\cos(x) \)[/tex] moves the entire range up by 7 units. Thus, [tex]\( -\cos(x) + 7 \)[/tex] will vary between 6 and 8.
2. Basic Characteristics:
- The period of [tex]\( \cos(x) \)[/tex] is [tex]\( 2\pi \)[/tex]. Given the transformation of flipping and shifting, the period remains [tex]\( 2\pi \)[/tex]. This means that the pattern of the wave repeats every [tex]\( 2\pi \)[/tex] units along the x-axis.
3. Finding when the Gum Returns to the Wall:
- The "wall" in this case corresponds to [tex]\( y = 7 \)[/tex].
- We need to find where the graph of [tex]\( y = -\cos(x) + 7 \)[/tex] intersects the line [tex]\( y = 7 \)[/tex].
4. Solving [tex]\( -\cos(x) + 7 = 7 \)[/tex]:
- Set [tex]\( y = -\cos(x) + 7 \)[/tex] equal to 7:
[tex]\[ -\cos(x) + 7 = 7 \][/tex]
Subtract 7 from both sides:
[tex]\[ -\cos(x) = 0 \][/tex]
Multiply both sides by -1:
[tex]\[ \cos(x) = 0 \][/tex]
- The cosine of [tex]\( x \)[/tex] equals 0 at:
[tex]\[ x = \frac{\pi}{2} + k\pi \quad \text{for integer } k \][/tex]
This is because [tex]\( \cos\left(\frac{\pi}{2}\right) = 0 \)[/tex], and cosine has a period of [tex]\( \pi \)[/tex] for subsequent zeros.
5. Identifying All Intersections within 60 Feet:
- We need to find all [tex]\( x \)[/tex] values where [tex]\( \cos(x) = 0 \)[/tex] and [tex]\( 0 \leq x \leq 60 \)[/tex]:
[tex]\[ x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots, \frac{\pi}{2} + n\pi \][/tex]
- The maximum value of [tex]\( n \)[/tex] where [tex]\( x \leq 60 \)[/tex]:
[tex]\[ \frac{\pi}{2} + n\pi \leq 60 \][/tex]
Substitute the [tex]\( \frac{\pi}{2} \)[/tex] to a common term:
[tex]\[ \pi(n + \frac{1}{2}) \leq 60 \][/tex]
Divide both sides by [tex]\( \pi \)[/tex]:
[tex]\[ n + \frac{1}{2} \leq \frac{60}{\pi} \][/tex]
Approximately (since [tex]\( \pi \approx 3.14159265359 \)[/tex]):
[tex]\[ n + \frac{1}{2} \leq 19.0986 \][/tex]
Subtract [tex]\( \frac{1}{2} \)[/tex] from both sides:
[tex]\[ n \leq 18.5986 \][/tex]
The largest integer [tex]\( n \leq 18.5986 \)[/tex] is 18.
Thus, [tex]\( n \)[/tex] can take on the values from 0 to 18 inclusive, giving us 19 points since it includes both 0 and 18.
So, the gum returns to the wall 19 times as it travels a distance of 60 feet.
