(a)

Length of the ladder as a function of is .

Solve the equation in the interval .

Consider .

The general solution of is , where is an integer.

.

If , then .

If , then .

The solution on the interval is

Therefore, the maximum happens when .

(b)

Substitue in .

Therefore, the length of the longest ladder is about .

(c)

The function is .

Draw the coordinate plane.

Use a graphing utility graph the function in the interval .

Graph :

Observe the above graph :

At the angle , that minimizes the length .

At , .

(d)

The result in part (b) is " the length of the longest ladder is about ".

The result in part (c) is " At the angle , that minimizes the length ".

Compare the above two results, both are same.

(a)

The solution of on the interval is .

(b)

The length of the longest ladder is about

.

(c)

Graph of in the interval :

At the angle , that minimizes the length .

(d)

The result in part (b) and the result in part (c) are same.