Final Problems

In a right triangle, if  is the length of the hypotenuse then by Pythagoras’ Theorem  where  and  are the lengths of the other two sides.

1.  A rope of 36 yards long will exactly reach from the top of a fort (built on one bank of the river) to the opposite bank of the river known to be 24 yard broad. Required the height of the fort.

Solution:  and  so that .

2.  The height of a tree growing in the centre of a circular island 44 feet in diameter, is 75 feet. A line stretched from the top of it over to the hither edge of the water is 256 feet. What is the breadth of the stream provided the land on each side be level?

Solution:  and  so that . The distance from the tree to the edge of the water is  feet. Therefore the breadth of the stream is  feet.

Comment: The problem and solution may have been copied directly from Pike’s book. To five decimal places . Both Pike and Macdonald give the square root as  so the answer was truncated at two decimal places rather than rounded to two places.

3.  Suppose a ladder 60 feet long be so planted as to reach a window 37 feet from the ground on one side of the street and without moving it at the foot will reach a window 23 feet high on the other side. I demand the width of the street.

Comment: Macdonald made a diagram that could be confusing to some. I have given a diagram that better describes the situation.

 Macdonald’s Diagram A Better Diagram

Solution: When the height  feet and the length of the ladder  then . When the height  feet and the length of the ladder  then . Therefore the width of the street is  feet.

4.  Two ships sail from the same port. One goes due north 45 leagues and the other due west 76 leagues. How far are they asunder?

Solution:  and  so that  leagues.