A geometry problem by Francois Tardieu

Geometry Level 3

Given that $E, D, C$ are on a straight line.
And $AB=HD=0.5\text{ m}$ , $BD=4\text{ m}$ .

Find the minimum angle of $x$ (in degrees) such that it minimizes the length $EC$ .

Give your answer to 2 decimal places.

1 solution

Francois Tardieu
Oct 16, 2016

We want to minimize the length: $EC= ED+DC$ and $0 < x < \dfrac{\pi}{2}$

in $\Delta DBC$

$\sin x = \dfrac{BD}{DC}$ , so we get, $DC=\dfrac{4}{\sin x }$

in $\Delta EHD$

$\cos x = \dfrac{HD}{ED}$ , so we get, $ED=\dfrac{0,5}{\cos x }$

$EC= f(x)=\dfrac{0,5}{\cos x } +\dfrac{4}{\sin x }$

$f'(x)=\dfrac{-0,5\sin x} {\cos^2 x }+\dfrac{4 \cos x} {\sin^2 x }= \dfrac{-0,5\sin^3 x+4\cos^3 x } {\cos^2 x \sin^2 x}$

$f'(x)=0 \Longleftrightarrow -0,5\sin^3 x+4\cos^3 x=0 \Longleftrightarrow 0,5\sin^3 x=4\cos^3 x \Longleftrightarrow \tan^3 x= \dfrac{4 } {0,5}=8=2^3$

We get, $\tan x= 2$

and, $x= \tan^{-1} 2$

$x\simeq 63,43^o$

Note that the problem states "minimize the length ED". Is this a typo?

Calvin Lin Staff - 4 years, 8 months ago

hi, yes it EC to minimize. need to fix the solution too at 63.43^o

have a good day

Francois Tardieu - 4 years, 8 months ago

Thanks. I've updated the problem and answer accordingly.

Calvin Lin Staff - 4 years, 8 months ago