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

- 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=(Adjacent)/(Hypotenuse) \\ \\ \theta=45\degree,Adjacent=4ft,\text{ Length of ladder}=L \\ \\ \cos45\degree=(4)/(L) \\ \\ \therefore L=(4)/(\cos45\degree)=(4)/((√(2))/(2))=4*(2)/(√(2))=(8)/(√(2))*(√(2))/(√(2))=4√(2) \end{gathered}`

Height of Wall:

`\begin{gathered} \tan\theta=(Opposite)/(Adjacent) \\ \\ \theta=45\degree,Opposite=w,Adjacent=4 \\ \\ \therefore\tan45\degree=(w)/(4) \\ \\ \therefore w=4\tan45\degree \\ \\ w=4ft \end{gathered}`

Final Answer

`\begin{gathered} \text{ The length of the ladder is }4√(2)ft \\ \\ \text{ The height of the wall is }4ft \end{gathered}`