Answer :
To construct a roller coaster that adheres to the given criteria, we need to design a function [tex]\( y \)[/tex] that meets the following conditions:
1. The roller coaster should swoop down through the [tex]\( x \)[/tex]-axis at [tex]\( x=300 \)[/tex].
2. The roller coaster should cross upward through the [tex]\( x \)[/tex]-axis at [tex]\( x=700 \)[/tex].
3. The roller coaster should end smoothly at [tex]\( x=1000 \)[/tex] with an exponent of 2.
Given the function
[tex]\[ y = -ax(x-500)(x-1000) \][/tex]
we will explore how this function satisfies the conditions at the specified points.
### Step-by-Step Solution:
1. Initial Rise and Swoop Down at [tex]\( x=300 \)[/tex]:
- The function should go through [tex]\( x=300 \)[/tex] correctly. By plugging [tex]\( x=300 \)[/tex] into the function:
[tex]\[ y = -a(300)(300-500)(300-1000) = -a \cdot 300 \cdot -200 \cdot -700 \][/tex]
[tex]\[ y = -a \cdot 300 \cdot 200 \cdot 700 = -42000000a \][/tex]
- Thus, at [tex]\( x=300 \)[/tex], we get [tex]\( y=-42000000a \)[/tex].
2. Crossing Upward at [tex]\( x=700 \)[/tex]:
- The function should cross the [tex]\( x \)[/tex]-axis at [tex]\( x=700 \)[/tex]. By plugging [tex]\( x=700 \)[/tex] into the function:
[tex]\[ y = -a(700)(700-500)(700-1000) = -a \cdot 700 \cdot 200 \cdot -300 \][/tex]
[tex]\[ y = -a \cdot 700 \cdot 200 \cdot -300 = 42000000a \][/tex]
- Thus, at [tex]\( x=700 \)[/tex], we get [tex]\( y=42000000a \)[/tex].
3. Smooth Ending at [tex]\( x=1000 \)[/tex]:
- The function should smoothly end at [tex]\( x=1000 \)[/tex] because the equation sets [tex]\( x=1000 \)[/tex] as a zero, making the function evaluate to zero smoothly. By plugging [tex]\( x=1000 \)[/tex] into the function:
[tex]\[ y = -a(1000)(1000-500)(1000-1000) = -a \cdot 1000 \cdot 500 \cdot 0 \][/tex]
[tex]\[ y = -a \cdot 1000 \cdot 500 \cdot 0 = 0 \][/tex]
- Thus, at [tex]\( x=1000 \)[/tex], we get [tex]\( y=0 \)[/tex].
### Final Coaster Values:
- For [tex]\( x=0 \)[/tex]:
[tex]\[ y = -a(0)(0-500)(0-1000) = 0 \][/tex]
- So at [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex].
- Summary of Values:
- At [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex]
- At [tex]\( x=300 \)[/tex], [tex]\( y=-42000000a \)[/tex]
- At [tex]\( x=700 \)[/tex], [tex]\( y=42000000a \)[/tex]
- At [tex]\( x=1000 \)[/tex], [tex]\( y=0 \)[/tex]
Finally, assuming [tex]\( a = 1 \)[/tex] for simplicity:
- At [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex]
- At [tex]\( x=300 \)[/tex], [tex]\( y=-42000000 \)[/tex]
- At [tex]\( x=700 \)[/tex], [tex]\( y=42000000 \)[/tex]
- At [tex]\( x=1000 \)[/tex], [tex]\( y=0 \)[/tex]
Thus, the specific calculations confirm the roller coaster’s journey through the specified points with a swoop down at [tex]\( x=300 \)[/tex], crossing upward at [tex]\( x=700 \)[/tex], and ending smoothly at [tex]\( x=1000 \)[/tex].
1. The roller coaster should swoop down through the [tex]\( x \)[/tex]-axis at [tex]\( x=300 \)[/tex].
2. The roller coaster should cross upward through the [tex]\( x \)[/tex]-axis at [tex]\( x=700 \)[/tex].
3. The roller coaster should end smoothly at [tex]\( x=1000 \)[/tex] with an exponent of 2.
Given the function
[tex]\[ y = -ax(x-500)(x-1000) \][/tex]
we will explore how this function satisfies the conditions at the specified points.
### Step-by-Step Solution:
1. Initial Rise and Swoop Down at [tex]\( x=300 \)[/tex]:
- The function should go through [tex]\( x=300 \)[/tex] correctly. By plugging [tex]\( x=300 \)[/tex] into the function:
[tex]\[ y = -a(300)(300-500)(300-1000) = -a \cdot 300 \cdot -200 \cdot -700 \][/tex]
[tex]\[ y = -a \cdot 300 \cdot 200 \cdot 700 = -42000000a \][/tex]
- Thus, at [tex]\( x=300 \)[/tex], we get [tex]\( y=-42000000a \)[/tex].
2. Crossing Upward at [tex]\( x=700 \)[/tex]:
- The function should cross the [tex]\( x \)[/tex]-axis at [tex]\( x=700 \)[/tex]. By plugging [tex]\( x=700 \)[/tex] into the function:
[tex]\[ y = -a(700)(700-500)(700-1000) = -a \cdot 700 \cdot 200 \cdot -300 \][/tex]
[tex]\[ y = -a \cdot 700 \cdot 200 \cdot -300 = 42000000a \][/tex]
- Thus, at [tex]\( x=700 \)[/tex], we get [tex]\( y=42000000a \)[/tex].
3. Smooth Ending at [tex]\( x=1000 \)[/tex]:
- The function should smoothly end at [tex]\( x=1000 \)[/tex] because the equation sets [tex]\( x=1000 \)[/tex] as a zero, making the function evaluate to zero smoothly. By plugging [tex]\( x=1000 \)[/tex] into the function:
[tex]\[ y = -a(1000)(1000-500)(1000-1000) = -a \cdot 1000 \cdot 500 \cdot 0 \][/tex]
[tex]\[ y = -a \cdot 1000 \cdot 500 \cdot 0 = 0 \][/tex]
- Thus, at [tex]\( x=1000 \)[/tex], we get [tex]\( y=0 \)[/tex].
### Final Coaster Values:
- For [tex]\( x=0 \)[/tex]:
[tex]\[ y = -a(0)(0-500)(0-1000) = 0 \][/tex]
- So at [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex].
- Summary of Values:
- At [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex]
- At [tex]\( x=300 \)[/tex], [tex]\( y=-42000000a \)[/tex]
- At [tex]\( x=700 \)[/tex], [tex]\( y=42000000a \)[/tex]
- At [tex]\( x=1000 \)[/tex], [tex]\( y=0 \)[/tex]
Finally, assuming [tex]\( a = 1 \)[/tex] for simplicity:
- At [tex]\( x=0 \)[/tex], [tex]\( y=0 \)[/tex]
- At [tex]\( x=300 \)[/tex], [tex]\( y=-42000000 \)[/tex]
- At [tex]\( x=700 \)[/tex], [tex]\( y=42000000 \)[/tex]
- At [tex]\( x=1000 \)[/tex], [tex]\( y=0 \)[/tex]
Thus, the specific calculations confirm the roller coaster’s journey through the specified points with a swoop down at [tex]\( x=300 \)[/tex], crossing upward at [tex]\( x=700 \)[/tex], and ending smoothly at [tex]\( x=1000 \)[/tex].
