Chapter 7.4, Problem 32WE

To determine

To find: the value of x which is distance above the ground where the ropes cross.

Answer to Problem 32WE

The value of x is 4.

Explanation of Solution

Given:

The triangle dimension is shown below:

  

Calculation:

In triangles FCB and EDB,

  $\begin{array}{l}\angle D=\angle C=90°\\ \angle B=\angle B\left(\text{Common}\right)\\ \therefore \Delta FCB\sim \Delta EDB\left(AA\right)\end{array}$

Hence,

  $\begin{array}{l}\frac{FC}{ED}=\frac{CB}{DB}\\ \frac{6}{x}=\frac{y+z}{z}\\ \frac{x}{6}=\frac{z}{y+z}\mathrm{......}\left(1\right)\end{array}$

In triangles ABC and EDC,

  $\begin{array}{l}\angle D=\angle B=90°\\ \angle C=\angle C\left(\text{Common}\right)\\ \therefore \Delta ABC\sim \Delta EDC\left(AA\right)\end{array}$

Hence,

  $\begin{array}{l}\frac{AB}{ED}=\frac{CB}{DC}\\ \frac{12}{x}=\frac{y+z}{y}\\ \frac{x}{12}=\frac{y}{y+z}\mathrm{......}\left(2\right)\end{array}$

Add equations (1) and (2) as follows:

  $\begin{array}{l}\frac{x}{6}+\frac{x}{12}=\frac{z}{y+z}+\frac{y}{y+z}\\ \frac{2x+x}{12}=\frac{y+z}{y+z}\\ \frac{3x}{12}=1\\ x=4\end{array}$

Thus, thevalue of xis 4.