Question

Help with math question precalculus A city averages 14 hours of daylight in June (the longest days) and 10 in December (the shortest days). Assume that the number of hours of daylight varies sinusoidal over a period of twelve months. Use the data above to find the following:Amplitude -Vertical shift-B value-Period- Choices Pi/12210Up 1212 months Pi/68Y=2sin(pi/2 x)+12Pi/84Y=12 sin(pi/4 x)+10

Answer 1

Given that:

Maximum hours = 14 hours

Minimum hours = 10 hours

Period = 12

Find amplitude.

$\begin{gathered} \text{Amplitude, a =}\frac{\max -\min }{2} \\ =\frac{14-10}{2} \\ =\frac{4}{2} \\ =2 \end{gathered}$

Find vertical shift, k.

$\begin{gathered} k=\frac{\max +\min }{2} \\ =\frac{14+10}{2} \\ =\frac{24}{2} \\ =12 \end{gathered}$

Find b from the formula

$P\text{eriod}=\frac{2\pi}{b}$$\begin{gathered} 12=\frac{2\pi}{b} \\ b=\frac{2\pi}{12} \\ =\frac{\pi}{6} \end{gathered}$

The june month means x = 6 has maximum daylight hours. So, we need to shift he maximum at x = 6.

Then h = 3.

Plug the obtained values into the formula:

$y=a\sin (b(x-h))+k$

where y gives the number of daylight hours and x is the number of months since January.

$y=2\sin (\frac{\pi}{6}(x-3))+12$