Chapter 4.8, Problem 44E

Solution

a.

The period of the sleep-awake cycle of a newborn, a four-year-old and of an adult.

The period of the sleep-awake cycle of a newborn, a four-year-old and of an adult are 6 hours, 24 hours and 24 hours respectively.

Given:

Given, human sleep-awake cycles at three different ages are described by the following graphs:

  

The portions of the graphs above the horizontal lines represent times awake, and the portions below represent times asleep.

Calculation:

The period of a function is the distance after which the function repeats itself.

From the given cycles, for a newborn the cycle repeats itself every 6 hours.

And for a four-year-old and an adult, the cycle repeats every 24 hours.

The period of the sleep-awake cycle of a newborn, a four-year-old and of an adult are 6 hours, 24 hours and 24 hours respectively.

Conclusion:

The period of the sleep-awake cycle of a newborn, a four-year-old and of an adult are 6 hours, 24 hours and 24 hours respectively.

b.

The sleep-awake cycle among the three that is the closest to being modeled by a function $y=a\sin bx$

The sleep-awake cycle among the three that is the closest to being modeled by a function $y=a\sin bx$ is the sleep-awake cycle of the newborn.

Given:

Given, human sleep-awake cycles at three different ages are described by the following graphs:

  

The portions of the graphs above the horizontal lines represent times awake, and the portions below represent times asleep.

Calculation:

From the given cycles, it can be seen that the cycle follows the pattern of fixed width between maximum and minimum only for a newborn. Hence the sleep-awake cycle among the three that is the closest to being modeled by a function $y=a\sin bx$ is the sleep-awake cycle of the newborn.

Conclusion:

The sleep-awake cycle among the three that is the closest to being modeled by a function $y=a\sin bx$ is the sleep-awake cycle of the newborn.