Let's break down the question and provide the required equations step-by-step.
### Ascent Up the Tower:
The ascent of the riders up the tower is a linear relationship. The height, [tex]\( h \)[/tex], is equal to the ascent rate multiplied by the time, [tex]\( t \)[/tex].
Given data:
- The ascent rate is [tex]\( 20 \)[/tex] feet per second.
Therefore, the equation representing the ascent is:
[tex]\[ h = 20t \][/tex]
### Free Fall Down the Tower:
The free fall down the tower is a quadratic relationship. The distance from the top to the bottom of the free fall, [tex]\( d \)[/tex], is given by the equation:
[tex]\[ d = -16t^2 + d_0 \][/tex]
where:
- [tex]\( d_0 \)[/tex] is the initial distance above the bottom of the free fall.
Given data:
- The total height of the tower is [tex]\( 350 \)[/tex] feet.
- The free fall covers [tex]\( 288 \)[/tex] feet.
- Thus, the initial height before free fall, [tex]\( d_0 \)[/tex], is calculated as:
[tex]\[ d_0 = 350 - 288 = 62 \text{ feet} \][/tex]
Therefore, the equation representing the free fall is:
[tex]\[ d = -16t^2 + 62 \][/tex]
### Final Equations:
1. The equation for the ascent (linear relationship):
[tex]\[ h = 20t \][/tex]
2. The equation for the free fall (quadratic relationship):
[tex]\[ d = -16t^2 + 62 \][/tex]
These are the required equations representing the relationships for the ride at the amusement park.
