A bee on the bowl walks around the circle of radius 4 centered at the origin. What is the maximum and Minimum hotness felt by the bee?
How do I find the maxima and minima on the circle's border. I know that the absolute maxima and minima of the temperature profile are infinity and 0. But how do I find it specific to the border of the circle?
Parametrically, the path of the bee is
$(x(\theta), y(\theta)) = ( 4 \cos \theta, 4 \sin \theta ) $
Substitute that into the temperature of the plate
$T(\theta) = 80 \cos^2 \theta + 16 \sin^2 \theta - 48 \cos \theta \sin \theta $
Using the trigonometric identities $ \cos^2 \theta = \dfrac{1 + \cos(2\theta)}{2} , \sin^2 \theta = \dfrac{1 - \cos(2 \theta)}{2}, \sin(2 \theta) = 2 \cos \theta \sin \theta $, the above expression becomes
$T(\theta) = 48 + 32 \cos(2 \theta) - 24 \sin(2 \theta) $
Hence,
$T_{Max} = 48 + \sqrt{ 32^2 + 24^2 } = 48 + 40 = 88 $
$T_{Min} = 48 - 40 = 8 $