Answer :
Certainly! Let's break this problem into two parts as described:
### Part 1: Trip Up the Tower
The height [tex]\( h \)[/tex] of the riders as they travel up the tower is given by a linear relationship. The rate of travel is constant at 20 feet per second. The height can be modeled as a function of time [tex]\( t \)[/tex]:
[tex]\[ h = 20t \][/tex]
Here, [tex]\( h \)[/tex] is the height in feet, and [tex]\( t \)[/tex] is the time in seconds.
### Part 2: Free Fall Down the Tower
For the descent, we have a quadratic relationship. The distance fallen [tex]\( d \)[/tex] can be modeled by the equation:
[tex]\[ d = -16t^2 + d_0 \][/tex]
In this equation:
- [tex]\( d \)[/tex] represents the distance fallen from the starting point of the free fall.
- [tex]\( t \)[/tex] is the time in seconds since the free fall began.
- [tex]\( d_0 \)[/tex] is the initial height from which the free fall starts.
Given that the initial height for the free fall ( [tex]\( d_0 \)[/tex] ) is 288 feet, the equation becomes:
[tex]\[ d = -16t^2 + 288 \][/tex]
So, to summarize:
- The equation representing the trip up the tower is: [tex]\( h = 20t \)[/tex]
- The equation representing the free fall down the tower is: [tex]\( d = -16t^2 + 288 \)[/tex]
These equations capture the height during the ascent and the distance fallen during the descent, respectively.
### Part 1: Trip Up the Tower
The height [tex]\( h \)[/tex] of the riders as they travel up the tower is given by a linear relationship. The rate of travel is constant at 20 feet per second. The height can be modeled as a function of time [tex]\( t \)[/tex]:
[tex]\[ h = 20t \][/tex]
Here, [tex]\( h \)[/tex] is the height in feet, and [tex]\( t \)[/tex] is the time in seconds.
### Part 2: Free Fall Down the Tower
For the descent, we have a quadratic relationship. The distance fallen [tex]\( d \)[/tex] can be modeled by the equation:
[tex]\[ d = -16t^2 + d_0 \][/tex]
In this equation:
- [tex]\( d \)[/tex] represents the distance fallen from the starting point of the free fall.
- [tex]\( t \)[/tex] is the time in seconds since the free fall began.
- [tex]\( d_0 \)[/tex] is the initial height from which the free fall starts.
Given that the initial height for the free fall ( [tex]\( d_0 \)[/tex] ) is 288 feet, the equation becomes:
[tex]\[ d = -16t^2 + 288 \][/tex]
So, to summarize:
- The equation representing the trip up the tower is: [tex]\( h = 20t \)[/tex]
- The equation representing the free fall down the tower is: [tex]\( d = -16t^2 + 288 \)[/tex]
These equations capture the height during the ascent and the distance fallen during the descent, respectively.