Answer to Question #275063 in Calculus for Hetal

Question #275063

n the right triangle ABC, AB = 2, BC = 4 and ED is a line parallel to AB. Find the



angle α = angle BAD which minimizes the distance L, where L = AD + ED



1
Expert's answer
2021-12-08T04:28:28-0500

Let BD=x.BD=x.


Right triangle ABDABD


BD=ABtanα,AD=ABcosαBD=AB\tan \alpha, AD=\dfrac{AB}{\cos \alpha}

Given AB=2,BC=4.AB = 2, BC = 4.

Substitute


x=2tanα,AD=2cosαx=2\tan \alpha, AD=\dfrac{2}{\cos \alpha}

Since EDAB,ED\parallel AB, then right triangles ABCABC and ABDABD are similar.


EDAB=CDBC=>ED=ABBCCD\dfrac{ED}{AB}=\dfrac{CD}{BC}=>ED=\dfrac{AB}{BC}\cdot CD

CD=BCBD=42tanαCD=BC-BD=4-2\tan \alpha

Substitute


ED=24(42tanα)=2tanαED=\dfrac{2}{4}(4-2\tan \alpha)=2-\tan \alpha

Then


L=AD+ED=2cosα+2tanα,L=AD+ED=\dfrac{2}{\cos \alpha}+2-\tan \alpha,0°<α<90°0\degree<\alpha<90\degree

Find the first derivative with respect to α\alpha


Lα=2cos2α(sinα)1cos2αL'_{\alpha}=-\dfrac{2}{\cos ^2 \alpha}(-\sin \alpha)-\dfrac{1}{\cos ^2 \alpha}

=2sinα1cos2α=\dfrac{2\sin \alpha-1}{\cos ^2 \alpha}

Find the critical number(s)


Lα=0=>2sinα1cos2α=0L'_{\alpha}=0=>\dfrac{2\sin \alpha-1}{\cos ^2 \alpha}=0

sinα=12,0°<α<90°\sin \alpha=\dfrac{1}{2}, 0\degree<\alpha<90\degree

The critical number is 30°.30\degree.

If 0°<α<30°,Lα<0,L0\degree<\alpha<30\degree, L'_{\alpha}<0, L decreases.

If 30°<α<90°,Lα>0,L30\degree<\alpha<90\degree, L'_{\alpha}>0, L increases.

The function LL has a local minimum at α=30°.\alpha=30\degree.

Since the function LL has the only extremum for 0°<α<90°,0\degree<\alpha<90\degree, then the function LL has the absolute minimum for 0°<α<90°0\degree<\alpha<90\degree at α=30°.\alpha=30\degree.


BAD=α=30°.\angle BAD=\alpha=30\degree.



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