(a) The given ladder positions are modified as above diagram

length of ladder

then length of ladder is calculated as

From triangle

\

 To draw graph for 

Substitute different values for θ in a given function then

    ( substitute  )

    ( substitute  )

\

      ( substitute  )

By substituting all the above values draw graph

(c) To find least value of the θ

apply derivative for 

As is constant    and  

                      ( simplify)

        ( simplify)

    ( simplify)

                       ( simplify)

                                 ( simplify)

\

(d) Here from first figure in step 1

The length of ladder is AE.

If we try to increase 1 ft length at side A of the ladder then 1 ft length will be decreased at side E of the ladder.

Length of ladder that can be carried around the corner is always constant.

So the length of longest ladder ( maximum ) that can be carried around the corner is same as the length of smallest ladder ( minimum ).

Length of longest ladder

         ( substitute )

         ( simplify )

                           ( simplify )

                                     ( simplify )

                                          ( simplify )

The solution is

(a)  Function is   is proved

(b) Graph for  Function is drawn

(c)  Least value of angle 

(d)  Length of longest ladder is