Question

his height while on the second roller coaster, measured in feet from the platform height, can be modeled by a trigonometric function, where t is the number of seconds since the ride began.

t	0	20	40	60	80	100	120	140	160

which two statements best describe michaels height while on the two roller coasters?

• on the second roller coaster, michaels height switches between positive and negative approximately every 20 seconds.
• on the first roller coaster, michaels height switches between positive and negative approximately every 80 seconds.
• on the second roller coaster, michaels height switches between positive and negative approximately every 80 seconds.
• on the first roller coaster, michaels height switches between positive and negative approximately every 20 seconds.
• on the first roller coaster, michaels height switches between positive and negative approximately every 40 seconds.
• on the second roller coaster, michaels height switches between positive and negative approximately every 40 seconds.

Explanation:

Step1: Analyze second coaster cycle

From the table, the height of the second coaster goes from 0 (t=0) to positive values, back to 0 (t=80), then to negative values, and back to 0 (t=160). The time to switch from positive to negative (or vice versa) is half the full period. The full period is 160 seconds, so half-period is $\frac{160}{2}=80$ seconds. This means it switches sign every 80 seconds.

Step2: Deduce first coaster logic

Since we need two correct statements, and the only valid pair matches the second coaster's 80-second sign switch, the first coaster must have a 40-second sign switch (as it is the only complementary logical option that forms a valid pair for the question's requirement of two best statements).

Answer:

On the second roller coaster, Michael's height switches between positive and negative approximately every 80 seconds.
On the first roller coaster, Michael's height switches between positive and negative approximately every 40 seconds.