Question

greg is in a car at the top of a roller - coaster ride. the distance, d, of the car from the ground as the car descends is determined by the equation $d = 144 - 16t^2$, where t is the number of seconds it takes the car to travel down to each point on the ride. for which interval of time is gregs car moving in the air?

• $0 < t \leq 3$
• $- 3 < t \leq 3$
• $t > 3$
• $0 < t < 3$

Explanation:

Step1: Understand the problem

The car is in the air when \( d>0 \) (distance from ground is positive, meaning it's not on the ground yet). The equation is \( d = 144 - 16t^2 \). So we need to solve \( 144 - 16t^2>0 \) for \( t \), considering time \( t\geq0 \) (since time can't be negative in this context).

Step2: Solve the inequality \( 144 - 16t^2>0 \)

First, factor the left - hand side:
\( 144-16t^{2}=16(9 - t^{2})=16(3 - t)(3 + t) \)
So the inequality becomes \( 16(3 - t)(3 + t)>0 \). Since \( 16>0 \), we can divide both sides by 16 without changing the inequality sign:
\( (3 - t)(3 + t)>0 \)
The roots of the equation \( (3 - t)(3 + t)=0 \) are \( t = 3 \) and \( t=-3 \).
We consider the intervals determined by these roots: \( (-\infty,-3) \), \( (-3,3) \), and \( (3,\infty) \).

• For \( t < - 3 \), let's take \( t=-4 \). Then \( (3-(-4))(3 + (-4))=(7)(-1)=-7<0 \), so the inequality is not satisfied.
• For \( - 3
• For \( t>3 \), let's take \( t = 4 \). Then \( (3 - 4)(3 + 4)=(-1)(7)=-7<0 \), so the inequality is not satisfied.

But since time \( t \) represents the number of seconds the car has been traveling down, \( t>0 \) (we start counting time when the car starts descending from the top). So the valid interval for \( t \) when \( d>0 \) (car is in the air) is \( 0 < t<3 \). We also need to check the endpoints: when \( t = 0 \), the car is at the top (just starting), and when \( t = 3 \), \( d=144-16\times9=144 - 144 = 0 \), which means the car is on the ground. So the car is in the air when \( 0 < t<3 \).

Answer:

D. \( 0 < t<3 \) (assuming the last option is D, if the options are labeled as A: \( 0 < t\leq3 \), B: \( - 3 3 \), D: \( 0 < t<3 \))