Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
