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Imagine being strapped into your seat at the bottom of a 350-foot tower, with your feet dangling just above the ground. You make the trip up the tower at a steady rate of 20 feet per second, stop at the top of the tower to hang for a few seconds, then suddenly drop in a free fall for 288 feet!

1. Linear Relationship:
The trip up the tower is a linear relationship. The height of the riders, [tex]$h$[/tex], is equal to the constant rate multiplied by the time, [tex]$t$[/tex], since they began the trip up.

Write an equation representing this relationship.

Answer: [tex]$h = 20t$[/tex]

2. Quadratic Relationship:
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 modeled by this equation, where [tex]$t$[/tex] is the time since the free fall began and [tex]$d_0$[/tex] is the initial distance above the bottom of the free fall.

[tex]$
d = -16t^2 + d_0
$[/tex]

Write an equation representing this relationship.

Answer: [tex]$d = -16t^2 + 288$[/tex]

Enter the correct answer in the box.



Answer :

GinnyAnswer
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.

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