The crossing ladders problem is the following: Two ladders of length and , with ≤ , are leaning across an alleyway between two buildings as shown in Figure 2.22. If they cross at a height , what is...

The crossing ladders problem is the following: Two ladders of length

and
, with

≤
, are leaning across an alleyway between two buildings as shown in Figure 2.22. If they cross at a height
, what is the width

of the alleyway?

(a) Using similar triangles and the Pythagorean theorem show that

²+
²
=

²,

²
+

²
=

², and

In these formulas,

and

are the vertical heights of the two ladders. From this show that the problem reduces to solving the following equation for
:

(b) Explain why

≤

≤
. Also, by sketching the functions in the equation from part (a), show that there are two positive solutions for

(assuming that


)Note that you might find it easier to first rewrite the equation before doing the sketch.

(c) Newton’s method is going to be used to find
. What does (2.10) reduce to in this case? Based on part (b), what would be a good choice for the starting point in this case? Make sure to explain why.

(d) The exact solution is easy to determine in the case of when

=
. For this case, picking a value for

and
, use Newton’s method to find
, and show that it gives the correct result.

(e) Taking

= 20,

= 30, and

= 8, use Newton’s method to compute

and from this determine
. Your answers should be correct to at least eight significant digits. Also, state what you used for a starting value, and explain why you made this choice.