Answer
The shortest day in Minneapolis is $~~8.6~hours$
The longest day in Minneapolis is $~~15.4~hours$
Work Step by Step
We can convert $44.88^{\circ}$ to units of radians:
$L = (44.88^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 0.7833~rad$
We can convert $-23.44^{\circ}$ to units of radians:
$D = (-23.44^{\circ})(\frac{\pi~rad}{180^{\circ}}) = -0.4091~rad$
We can calculate the shortest day in Minneapolis:
$cos(0.1309~H) = -tan~D~tan~L$
$cos(0.1309~H) = -tan~(-0.4091)~tan~(0.7833)$
$cos(0.1309~H) = 0.431746$
$0.1309~H = cos^{-1}(0.431746)$
$0.1309~H = 1.12436$
$H = \frac{1.12436}{0.1309}$
$H = 8.6~hours$
The shortest day in Minneapolis is $~~8.6~hours$
We can calculate the longest day in Minneapolis:
$cos(0.1309~H) = -tan~D~tan~L$
$cos(0.1309~H) = -tan~(0.4091)~tan~(0.7833)$
$cos(0.1309~H) = -0.431746$
$0.1309~H = cos^{-1}(-0.431746)$
$0.1309~H = 2.017224$
$H = \frac{2.017224}{0.1309}$
$H = 15.4~hours$
The longest day in Minneapolis is $~~15.4~hours$
