Question

mathematics ii a thayer fall 25 analyzing functional relationships a fountain sprays water into a pool as part of the filtration system. the projected path of water is modeled by the function given in the table, where x represents the time in seconds since the water left the fountain and f(x) represents the height in feet above the pools water line. how many seconds does the water travel through the air?

Explanation:

Step1: Understand the problem

We need to find the time the water travels through the air. The height \( f(x) \) is 0 when the water is at the pool's surface (starts or ends its air travel). Wait, actually, when the water is in the air, it starts at \( x = 0 \) (height 1.5) and ends when it hits the pool, but also, when does it come back? Wait, looking at the table:

\( x \): 0, 1, 2, 3, 4, 5

\( f(x) \): 1.5, 2, 1.5, 0, -2.5, -6

Wait, the water is in the air from when it leaves the fountain (x=0) until it hits the pool (f(x)=0 or below? Wait, no. Wait, the height is above the pool's water line. So when f(x)=0, it's at the pool's surface. Wait, at x=3, f(x)=0. But also, at x=0, f(x)=1.5 (leaving the fountain). Wait, maybe the time in air is from when it leaves (x=0) until it hits the pool (x=3)? No, wait, maybe we need to see when the height is non - negative? Wait, no, the question is "how many seconds does the water travel through the air". So the water is in the air from the time it leaves the fountain (x = 0) until it hits the pool (when f(x)=0 or when it goes below? Wait, no, when f(x)=0, it's at the pool's surface. Wait, at x = 3, f(x)=0. But also, at x=0, f(x)=1.5 (leaving the fountain). Wait, maybe the time is from x=0 to x=3? But let's check the values. Wait, maybe the water is in the air until it hits the pool, so when f(x)=0, that's when it hits. But also, maybe we need to see the time between the first and last time it's in the air. Wait, no, the function is a parabola - like (since it goes up to x=1, f(x)=2, then down). So the water is in the air from x=0 (leaving) until x=3 (hitting the pool), and also, wait, at x=5, it's at -6, which is below. Wait, no, the time in air is the time from when it leaves the fountain (x=0) until it hits the pool (x=3)? But wait, let's check the table again. Wait, maybe the question is about the time when the water is above the pool (f(x)≥0). Wait, at x=0, f(x)=1.5; x=1, 2; x=2, 1.5; x=3, 0; x=4, -2.5; x=5, -6. So the water is in the air from x=0 to x=3, because at x=3, it's at the pool's surface (f(x)=0). So the time is 3 - 0=3? No, wait, maybe I misread. Wait, the options are 1.5,5,3,-6. Wait, -6 is a height, not time. 1.5 is a height. 5 is x=5, 3 is x=3. Wait, when does the water start and end? At x=0, it's at 1.5 (leaving the fountain). At x=3, it's at 0 (hitting the pool). So the time in air is 3 seconds? Wait, but let's check the table again. Wait, maybe the water is in the air from the first time it leaves (x=0) until it hits the pool (x=3). So the duration is 3 - 0=3 seconds.

Step2: Confirm with the table

Looking at the table, the height \( f(x) \) is positive (or zero) from \( x = 0 \) to \( x = 3 \) (at \( x = 3 \), \( f(x)=0 \)). After \( x = 3 \), the height is negative (below the pool's water line), meaning the water has hit the pool. So the time the water travels through the air is from \( x = 0 \) to \( x = 3 \), which is \( 3-0 = 3 \) seconds.

Answer:

3