Question

select the correct answer.
two friends are sitting on a seesaw at the park. the height, ( h(t) ), of the left seat, in inches above the ground, is modeled by a sine function. let ( t ) represent the time, in seconds, since the friends started seesawing.

( t )	( h(t) )
1	48
2	24
3	0
4	24
5	48
6	24

if the friends start seesawing with the plank in a level position, how long does it take for the friend in the left seat to move upward, downward, and then back to the initial position?

a. 4 seconds
b. 6 seconds
c. 3 seconds
d. 2 seconds

Explanation:

Step1: Analyze the initial position

At \( t = 0 \), \( h(t)=24 \) inches. This is the initial height.

Step2: Track the motion

- From \( t = 0 \) to \( t = 1 \), height increases to 48 (upward).
- From \( t = 1 \) to \( t = 3 \), height decreases to 0 (downward, reaching the lowest point at \( t = 3 \)).
- From \( t = 3 \) to \( t = 6 \), height increases back to 24 (upward, back to initial position at \( t = 6 \))? Wait, no, wait the cycle: Wait, let's check the values again. Wait, at \( t = 0 \), \( h = 24 \); \( t = 1 \), \( h = 48 \); \( t = 2 \), \( h = 24 \); \( t = 3 \), \( h = 0 \); \( t = 4 \), \( h = 24 \); \( t = 5 \), \( h = 48 \); \( t = 6 \), \( h = 24 \). Wait, actually, the motion: starting at 24 (t=0), goes up to 48 (t=1), down to 24 (t=2), down to 0 (t=3), up to 24 (t=4), up to 48 (t=5), down to 24 (t=6). Wait, no, the cycle of moving upward, downward, and back to initial. Wait, the initial position is \( h = 24 \) at \( t = 0 \). Let's see the time to go from 24 (t=0), up to 48 (t=1), down to 24 (t=2), down to 0 (t=3), up to 24 (t=4), up to 48 (t=5), down to 24 (t=6). Wait, no, actually, the period here: from t=0 (24) to t=6 (24), that's 6 seconds? Wait, no, wait the question is: move upward, downward, and back to initial. Wait, initial position is \( h = 24 \) at \( t = 0 \). Let's track the motion:

- Upward: from \( t = 0 \) to \( t = 1 \) (h increases from 24 to 48).
- Downward: from \( t = 1 \) to \( t = 3 \) (h decreases from 48 to 0; wait, no, at \( t = 2 \), h is 24, so from t=1 (48) to t=2 (24) is downward, t=2 to t=3 (0) is more downward. Then upward: from t=3 (0) to t=4 (24) is upward, t=4 to t=5 (48) is upward, t=5 to t=6 (24) is downward? Wait, no, I think I messed up. Wait, the key is to find the time to go from initial (t=0, h=24), move up, down, and back to initial. Wait, looking at the table:

At t=0: 24 (initial)

t=1: 48 (up)

t=2: 24 (back to initial? No, wait 24 is same as t=0, but wait t=0 to t=2: up then down to initial? Wait, no, t=0:24, t=1:48 (up), t=2:24 (down to initial), t=3:0 (down further), t=4:24 (up to initial), t=5:48 (up), t=6:24 (down to initial). Wait, no, the cycle of moving up, down, and back to initial: from t=0 to t=6, but that's 6 seconds? Wait, no, wait the question is "move upward, downward, and then back to the initial position". Let's check the positions:

- Initial position: t=0, h=24.

- Upward: t=0 to t=1 (h increases to 48).

- Downward: t=1 to t=3 (h decreases to 0; wait, no, at t=2, h=24, which is same as initial. Wait, maybe my initial analysis is wrong. Wait, let's list the heights:

t: 0, h:24

t:1, h:48 (up)

t:2, h:24 (down to initial)

t:3, h:0 (down further)

t:4, h:24 (up to initial)

t:5, h:48 (up)

t:6, h:24 (down to initial)

Wait, so from t=0 (initial) to t=6 (back to initial), but that's 6 seconds? But wait, the options include 6 seconds (option B). Wait, but let's check the motion: upward (t=0 to t=1), downward (t=1 to t=3, but at t=2, it's back to initial? No, t=2, h=24, same as t=0. Wait, maybe the cycle is from t=0 to t=6, but the motion: up (0-1), down (1-3), up (3-6) back to initial? Wait, no, at t=6, h=24, same as t=0. So the time to go up, down, and back to initial is 6 seconds? Wait, but let's check the options. Option B is 6 seconds. Wait, but let's re-examine the table:

Wait, the problem says "move upward, downward, and then back to the initial position". Let's track the height:

- Start at t=0 (h=24, initial).

- Move upward: from t=0 to t=1 (h goes from 24 to 48).

- Move downward: from t=1 to t=3 (h goes from 48 to 0; at t=2, h=24, which is the initial height, but then continues down to 0 at t=3).

- Move upward: from t=3 to t=6 (h goes from 0 to 24 at t=4, then to 48 at t=5, then back to 24 at t=6). Wait, no, at t=6, h=24, which is initial. So the total time is from t=0 to t=6, which is 6 seconds. So the answer is 6 seconds (option B).

Answer:

B. 6 seconds