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Have you ever been on or seen a ride like this at a fair or amusement park? Imagine being strapped into your seat at the bottom of this 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!

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.

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 each relationship.
Enter the correct answer in the box.



Answer :

GinnyAnswer
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.