Question

multiple choice 1.4 points
the hours of daylight in atlanta oscillate according to this model where t is months from january 1st and h(t) is hours of daylight in a day. when should you plant a spring garden if you want to do it during a month where there are 10 hours of daylight? (hint: january is t = 0)
$h(t)=-2.25\cos(\frac{\pi}{6}t)+10$
february
march
april
may

Explanation:

Step1: Set up the equation

Set \(h(t) = 10\), so \(10=-2.25\cos(\frac{\pi}{6}t)+10\).

Step2: Simplify the equation

Subtract \(10\) from both sides: \(0=-2.25\cos(\frac{\pi}{6}t)\). Then \(\cos(\frac{\pi}{6}t) = 0\).

Step3: Solve for \(t\)

We know that \(\cos(x)=0\) when \(x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}\). So \(\frac{\pi}{6}t=\frac{\pi}{2}+k\pi\). Divide both sides by \(\pi\): \(\frac{1}{6}t=\frac{1}{2}+k\). Multiply both sides by \(6\): \(t = 3+6k\).
Since \(t\) represents months (\(t\geq0\) and \(t\) is an integer for months). When \(k = 0\), \(t=3\).

Answer:

March (because when \(t = 3\), it represents March as \(t = 0\) is January)