Question

Thaddeus models the number of hours of daylight in his town asD(t) 3sin(pi/6t) + 12, where D is the number of daylight hours and t is the time in months since January 1.What are the least and greatest numbers of daylight hours over the course of a year?A. Least: 7.5 hours; greatest: 15 hoursB. Least: 3 hours; greatest: 12 hoursC. Least: 9 hours; greatest: 15 hoursD. Least: 6 hours; greatest: 12 hours

Answer 1

Take the following expression into account:

$D(t)=3\sin (\frac{\pi}{6}t)+12$

Since the term:

$3\sin (\frac{\pi}{6}t)$

Is between -3 and 3 depending on the specific value of t, we know that:

[tex]\begin{gathered} -3<3\sin (\frac{\pi}{6}t)<3 \\ \Rightarrow-3+12<3\sin (\frac{\pi}{6}t)+12<3+12 \\ \Rightarrow9<3\sin (\frac{\pi}{6}t)+12<15 \\ \Rightarrow9Therefore, the hours of daylight range from at least 9 hours and at most 15 hours.