A wall 10 ft high along the property line is 8 ft from a house.

For shortest ladder
$L^{2/3} = x^{2/3} + y^{2/3}$
 

 

$L^{2/3} = 8^{2/3} + 10^{2/3}$

$L = 25.4 ~ \text{ft}$

$a = 10\csc \theta$

$b = 8\sec \theta$
 

Length of ladder
$L = a + b$

$L = 10\csc \theta + 8\sec \theta$
 

For shortest ladder
$\dfrac{dL}{d\theta} = -10\csc \theta \cot \theta + 8\sec \theta \tan \theta = 0$

$8\sec \theta \tan \theta = 10\csc \theta \cot \theta$

$8 \cdot \dfrac{1}{\cos \theta} \cdot \dfrac{\sin \theta}{\cos \theta} = 10 \cdot \dfrac{1}{\sin \theta} \cdot \dfrac{\cos \theta}{\sin \theta}$

$\dfrac{\sin^3 \theta}{\cos^3 \theta} = \dfrac{10}{8}$

$\tan^3 \theta = \frac{5}{4}$

$\tan \theta = \dfrac{\sqrt[3]{5}}{\sqrt[3]{4}}$
 

 

From the $\theta$ in the figure above

$\csc \theta = \dfrac{\sqrt{4^{2/3} + 5^{2/3}}}{\sqrt[3]{5}}$

$\sec \theta = \dfrac{\sqrt{4^{2/3} + 5^{2/3}}}{\sqrt[3]{4}}$

 

Minimum length of $L$
$L_{min} = 10 \cdot \dfrac{\sqrt{4^{2/3} + 5^{2/3}}}{\sqrt[3]{5}} + 8 \cdot \dfrac{\sqrt{4^{2/3} + 5^{2/3}}}{\sqrt[3]{4}}$

$L_{min} = 25.4 ~ \text{ft}$