Question

A rooferprops a ladder against a wall so that thebottom of the ladder is 4 feet away from thewall. If the angle of elevation from the bottomof the ladder to the wall is 45°, how long is theladder? How tall is the wall?

Answer 1

Solution

- The sketc of the illustraction given in the question is given below:

- In order to solve ths question, we need to apply tSOHCAHTOA as the orientaton of the ladder and wall toform a right-angledtriangle

Length of the Ladder:

$\begin{gathered} \cos\theta=\frac{Adjacent}{Hypotenuse} \\ \\ \theta=45\degree,Adjacent=4ft,\text{ Length of ladder}=L \\ \\ \cos45\degree=\frac{4}{L} \\ \\ \therefore L=\frac{4}{\cos45\degree}=\frac{4}{\frac{\sqrt{2}}{2}}=4\times\frac{2}{\sqrt{2}}=\frac{8}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=4\sqrt{2} \end{gathered}$

Height of Wall:

$\begin{gathered} \tan\theta=\frac{Opposite}{Adjacent} \\ \\ \theta=45\degree,Opposite=w,Adjacent=4 \\ \\ \therefore\tan45\degree=\frac{w}{4} \\ \\ \therefore w=4\tan45\degree \\ \\ w=4ft \end{gathered}$

Final Answer

$\begin{gathered} \text{ The length of the ladder is }4\sqrt{2}ft \\ \\ \text{ The height of the wall is }4ft \end{gathered}$