# Problem of the Week Problem C and Solution Climbing Up and Down

## Problem

An obstacle course is created as follows. Two posts, the first being 4 m high and the second being 7.5 m high, are placed 12 m apart. A rope ladder starts on the ground, $$3$$ m from the base of the first post, and finishes at the top of the first post. A second rope ladder connects the tops of the two posts. A third rope ladder starts at the top of the second post and finishes on the ground $$4$$ m from the base of the second post. An illustration of the obstacle course is provided below.

To complete the obstacle course, Jesse has to climb along the three rope ladders. If each of the three rope ladders forms a straight line, then determine the total distance Jesse must travel on the rope ladders.

Note: You may find the following useful:

The Pythagorean Theorem states, “In a right-angled triangle, the square of the length of hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides.”

In the right-angled triangle shown, $$c$$ is the hypotenuse, $$a$$ and $$b$$ are the lengths of the other two sides, and $$c^2=a^2+b^2$$.

## Solution

We begin by labelling the diagram.
Let $$P$$ be where the first rope ladder meets the ground.
Let $$U$$ be the base of the first post and $$Q$$ be the top of the first post.
Let $$T$$ be the base of the second post and $$R$$ be the top of the second post.
Let $$S$$ be where the third rope ladder meets the ground.

The total distance Jesse must travel on the rope ladders is equal to $$PQ+QR+RS$$.

First, we will calculate the lengths of $$PQ$$ and $$RS$$.

Since $$\angle PUQ= 90\degree$$, we can apply the Pythagorean Theorem in $$\triangle PUQ$$. Thus, $$PQ^2 = QU^2 + PU^2 = 4^2 + 3^2 = 16+ 9 = 25$$. Therefore, $$PQ = 5$$, since $$PQ>0$$.

Similarly, $$\angle RTS= 90\degree$$, so we can apply the Pythagorean Theorem in $$\triangle RTS$$. Thus, $$RS^2 = RT^2 + TS^2 = 7.5^2 + 4^2 = 56.25 + 16 = 72.25$$. Therefore, $$RS = 8.5$$, since $$RS>0$$.

Now we will calculate the length of $$QR$$. Draw a line from $$Q$$ perpendicular to $$RT$$. Let $$A$$ be the point of intersection of the perpendicular with $$RT$$. Since $$QA$$ is perpendicular to $$RT$$, $$QATU$$ is a rectangle. Therefore, $$QA = UT = 12$$ and $$AT = QU = 4$$. Thus, $$AR = RT - AT = 7.5 - 4 = 3.5$$.

Since $$\angle QAR= 90\degree$$, we can apply the Pythagorean Theorem in $$\triangle QAR$$. Thus, $$QR^2 = QA^2 + AR^2 = 12^2 + 3.5^2 = 144 + 12.25 = 156.25$$. Therefore, $$QR = 12.5$$, since $$QR>0$$.

Therefore, the total distance Jesse must travel on the rope ladders is $$PQ + QR + RS = 5 + 12.5 + 8.5 = 26$$ m.